WEBVTT Kind: captions Language: en 00:00:01.160 --> 00:00:06.690 align:start position:0% [Music] 00:00:06.690 --> 00:00:06.700 align:start position:0% 00:00:06.700 --> 00:00:09.750 align:start position:0% [Music] 00:00:09.750 --> 00:00:09.760 align:start position:0% [Music] 00:00:09.760 --> 00:00:13.470 align:start position:0% [Music] hello<00:00:10.759> um<00:00:11.440> so<00:00:11.719> I'm<00:00:12.120> Phillip<00:00:12.799> uh<00:00:13.080> work<00:00:13.240> on<00:00:13.360> the 00:00:13.470 --> 00:00:13.480 align:start position:0% hello um so I'm Phillip uh work on the 00:00:13.480 --> 00:00:17.230 align:start position:0% hello um so I'm Phillip uh work on the mining<00:00:13.920> VM<00:00:14.920> and<00:00:15.640> um<00:00:16.640> yeah<00:00:16.800> today<00:00:17.000> we're<00:00:17.119> going 00:00:17.230 --> 00:00:17.240 align:start position:0% mining VM and um yeah today we're going 00:00:17.240 --> 00:00:18.550 align:start position:0% mining VM and um yeah today we're going to<00:00:17.400> talk<00:00:17.560> about<00:00:17.760> our<00:00:18.000> lessons<00:00:18.359> from 00:00:18.550 --> 00:00:18.560 align:start position:0% to talk about our lessons from 00:00:18.560 --> 00:00:21.830 align:start position:0% to talk about our lessons from integrating<00:00:19.279> uh<00:00:19.400> log<00:00:19.640> up<00:00:19.760> gkr<00:00:20.119> and<00:00:20.240> the<00:00:20.320> M<00:00:20.840> VM 00:00:21.830 --> 00:00:21.840 align:start position:0% integrating uh log up gkr and the M VM 00:00:21.840 --> 00:00:24.429 align:start position:0% integrating uh log up gkr and the M VM uh<00:00:22.240> it's<00:00:22.400> going<00:00:22.480> to<00:00:22.600> be<00:00:22.720> fairly<00:00:23.080> high<00:00:23.279> level<00:00:23.840> so 00:00:24.429 --> 00:00:24.439 align:start position:0% uh it's going to be fairly high level so 00:00:24.439 --> 00:00:27.910 align:start position:0% uh it's going to be fairly high level so I<00:00:24.640> wrote<00:00:25.000> a<00:00:25.199> little<00:00:25.439> hackmd<00:00:26.039> book<00:00:26.880> at 00:00:27.910 --> 00:00:27.920 align:start position:0% I wrote a little hackmd book at 00:00:27.920 --> 00:00:31.470 align:start position:0% I wrote a little hackmd book at bit.ly<00:00:28.920> log<00:00:29.119> up-gk<00:00:30.080> R<00:00:30.720> where<00:00:31.039> a<00:00:31.119> lot<00:00:31.240> of<00:00:31.359> the 00:00:31.470 --> 00:00:31.480 align:start position:0% bit.ly log up-gk R where a lot of the 00:00:31.480 --> 00:00:33.229 align:start position:0% bit.ly log up-gk R where a lot of the details<00:00:31.840> that<00:00:31.960> we<00:00:32.040> don't<00:00:32.239> cover<00:00:32.599> here<00:00:32.840> are<00:00:33.079> in 00:00:33.229 --> 00:00:33.239 align:start position:0% details that we don't cover here are in 00:00:33.239 --> 00:00:34.630 align:start position:0% details that we don't cover here are in that 00:00:34.630 --> 00:00:34.640 align:start position:0% that 00:00:34.640 --> 00:00:39.310 align:start position:0% that book<00:00:35.640> so<00:00:36.120> for<00:00:36.360> context<00:00:36.840> log<00:00:37.079> up<00:00:37.239> gkr<00:00:37.800> is<00:00:38.040> a<00:00:39.040> uh 00:00:39.310 --> 00:00:39.320 align:start position:0% book so for context log up gkr is a uh 00:00:39.320 --> 00:00:41.750 align:start position:0% book so for context log up gkr is a uh technique<00:00:39.879> proposed<00:00:40.320> by<00:00:40.719> Shahar<00:00:41.120> and<00:00:41.239> olrick 00:00:41.750 --> 00:00:41.760 align:start position:0% technique proposed by Shahar and olrick 00:00:41.760 --> 00:00:43.670 align:start position:0% technique proposed by Shahar and olrick around<00:00:42.039> this<00:00:42.200> time<00:00:42.440> last<00:00:42.640> year<00:00:43.320> in<00:00:43.440> their 00:00:43.670 --> 00:00:43.680 align:start position:0% around this time last year in their 00:00:43.680 --> 00:00:45.590 align:start position:0% around this time last year in their paper<00:00:44.200> improving<00:00:44.680> logarithmic<00:00:45.200> derivative 00:00:45.590 --> 00:00:45.600 align:start position:0% paper improving logarithmic derivative 00:00:45.600 --> 00:00:48.549 align:start position:0% paper improving logarithmic derivative lookups<00:00:46.320> using 00:00:48.549 --> 00:00:48.559 align:start position:0% lookups using 00:00:48.559 --> 00:00:52.590 align:start position:0% lookups using gkr<00:00:49.559> so<00:00:50.079> in<00:00:50.239> this<00:00:50.440> talk<00:00:50.920> we're<00:00:51.120> going<00:00:51.199> to<00:00:52.120> first 00:00:52.590 --> 00:00:52.600 align:start position:0% gkr so in this talk we're going to first 00:00:52.600 --> 00:00:55.349 align:start position:0% gkr so in this talk we're going to first look<00:00:52.760> at<00:00:53.079> what<00:00:53.239> log<00:00:53.440> up<00:00:53.680> is<00:00:54.079> first<00:00:54.239> of<00:00:54.399> all<00:00:55.160> and 00:00:55.349 --> 00:00:55.359 align:start position:0% look at what log up is first of all and 00:00:55.359 --> 00:00:58.069 align:start position:0% look at what log up is first of all and then<00:00:55.960> uh<00:00:56.520> how<00:00:56.800> log<00:00:57.039> up<00:00:57.239> is<00:00:57.399> typically<00:00:57.719> proved 00:00:58.069 --> 00:00:58.079 align:start position:0% then uh how log up is typically proved 00:00:58.079 --> 00:01:00.750 align:start position:0% then uh how log up is typically proved with<00:00:58.640> Starks<00:00:59.640> and<00:00:59.760> then<00:01:00.160> we'll<00:01:00.320> look<00:01:00.440> at<00:01:00.640> how 00:01:00.750 --> 00:01:00.760 align:start position:0% with Starks and then we'll look at how 00:01:00.760 --> 00:01:03.549 align:start position:0% with Starks and then we'll look at how you<00:01:00.879> do<00:01:01.039> it<00:01:01.199> with<00:01:01.359> gkr<00:01:02.000> which<00:01:02.120> is<00:01:02.280> the<00:01:02.440> main<00:01:03.440> the 00:01:03.549 --> 00:01:03.559 align:start position:0% you do it with gkr which is the main the 00:01:03.559 --> 00:01:05.590 align:start position:0% you do it with gkr which is the main the main<00:01:03.800> purpose<00:01:04.040> of<00:01:04.159> the<00:01:04.320> talk<00:01:05.199> then<00:01:05.360> our 00:01:05.590 --> 00:01:05.600 align:start position:0% main purpose of the talk then our 00:01:05.600 --> 00:01:09.109 align:start position:0% main purpose of the talk then our takeaway<00:01:05.920> is<00:01:06.000> from<00:01:06.200> doing 00:01:09.109 --> 00:01:09.119 align:start position:0% 00:01:09.119 --> 00:01:12.789 align:start position:0% so<00:01:10.119> so<00:01:10.720> really<00:01:11.000> quickly<00:01:11.560> log<00:01:11.840> up<00:01:12.119> is<00:01:12.320> a 00:01:12.789 --> 00:01:12.799 align:start position:0% so so really quickly log up is a 00:01:12.799 --> 00:01:14.630 align:start position:0% so so really quickly log up is a protocol<00:01:13.200> to<00:01:13.360> prove<00:01:13.640> the<00:01:13.840> equality<00:01:14.280> of<00:01:14.439> two 00:01:14.630 --> 00:01:14.640 align:start position:0% protocol to prove the equality of two 00:01:14.640 --> 00:01:18.590 align:start position:0% protocol to prove the equality of two multi<00:01:15.200> sets<00:01:16.200> uh<00:01:16.759> multiset<00:01:17.439> is<00:01:17.640> a<00:01:17.920> set<00:01:18.240> where 00:01:18.590 --> 00:01:18.600 align:start position:0% multi sets uh multiset is a set where 00:01:18.600 --> 00:01:20.910 align:start position:0% multi sets uh multiset is a set where repetition<00:01:19.159> is 00:01:20.910 --> 00:01:20.920 align:start position:0% repetition is 00:01:20.920 --> 00:01:23.950 align:start position:0% repetition is allowed<00:01:21.920> so<00:01:22.240> they're<00:01:22.680> super<00:01:23.040> super<00:01:23.320> useful<00:01:23.759> in 00:01:23.950 --> 00:01:23.960 align:start position:0% allowed so they're super super useful in 00:01:23.960 --> 00:01:26.230 align:start position:0% allowed so they're super super useful in Virtual<00:01:24.280> machines<00:01:25.159> uh<00:01:25.320> they<00:01:25.439> allow<00:01:25.680> you<00:01:25.840> to 00:01:26.230 --> 00:01:26.240 align:start position:0% Virtual machines uh they allow you to 00:01:26.240 --> 00:01:30.190 align:start position:0% Virtual machines uh they allow you to have<00:01:26.479> practically<00:01:26.960> unlimited<00:01:27.600> stack<00:01:28.439> def<00:01:29.439> um 00:01:30.190 --> 00:01:30.200 align:start position:0% have practically unlimited stack def um 00:01:30.200 --> 00:01:33.230 align:start position:0% have practically unlimited stack def um do<00:01:30.520> large<00:01:30.840> number<00:01:31.200> of<00:01:31.360> range<00:01:31.720> checks<00:01:32.399> so 00:01:33.230 --> 00:01:33.240 align:start position:0% do large number of range checks so 00:01:33.240 --> 00:01:34.950 align:start position:0% do large number of range checks so making<00:01:33.520> sure<00:01:33.759> that<00:01:33.960> a<00:01:34.159> given<00:01:34.479> value<00:01:34.720> in<00:01:34.799> your 00:01:34.950 --> 00:01:34.960 align:start position:0% making sure that a given value in your 00:01:34.960 --> 00:01:36.710 align:start position:0% making sure that a given value in your Trace<00:01:35.520> is<00:01:35.759> within<00:01:36.000> some 00:01:36.710 --> 00:01:36.720 align:start position:0% Trace is within some 00:01:36.720 --> 00:01:39.469 align:start position:0% Trace is within some bounds<00:01:37.720> and<00:01:38.280> uh<00:01:38.399> they<00:01:38.520> allow<00:01:38.840> communication 00:01:39.469 --> 00:01:39.479 align:start position:0% bounds and uh they allow communication 00:01:39.479 --> 00:01:41.749 align:start position:0% bounds and uh they allow communication between<00:01:39.840> different<00:01:40.119> parts<00:01:40.320> of<00:01:40.399> the<00:01:40.600> trace<00:01:41.600> so 00:01:41.749 --> 00:01:41.759 align:start position:0% between different parts of the trace so 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somewhere else in your trace the multi set<00:01:48.280> check<00:01:48.560> is<00:01:48.680> what<00:01:48.799> you<00:01:48.880> would<00:01:49.040> use<00:01:49.240> to 00:01:49.389 --> 00:01:49.399 align:start position:0% set check is what you would use to 00:01:49.399 --> 00:01:52.109 align:start position:0% set check is what you would use to connect<00:01:49.680> the<00:01:49.759> two<00:01:49.920> parts<00:01:50.119> of<00:01:50.240> the 00:01:52.109 --> 00:01:52.119 align:start position:0% connect the two parts of the 00:01:52.119 --> 00:01:55.270 align:start position:0% connect the two parts of the trace<00:01:53.119> so<00:01:53.360> for<00:01:53.560> example<00:01:54.079> this<00:01:54.200> is<00:01:54.439> a<00:01:55.040> uh 00:01:55.270 --> 00:01:55.280 align:start position:0% trace so for example this is a uh 00:01:55.280 --> 00:01:59.149 align:start position:0% trace so for example this is a uh depiction<00:01:55.840> of<00:01:56.119> the<00:01:56.960> uh<00:01:57.119> M<00:01:57.399> VM<00:01:57.719> Trace<00:01:58.479> so<00:01:58.680> a<00:01:58.799> 2d 00:01:59.149 --> 00:01:59.159 align:start position:0% depiction of the uh M VM Trace so a 2d 00:01:59.159 --> 00:02:00.510 align:start position:0% depiction of the uh M VM Trace so a 2d grid<00:01:59.399> of<00:01:59.520> number 00:02:00.510 --> 00:02:00.520 align:start position:0% grid of number 00:02:00.520 --> 00:02:04.510 align:start position:0% grid of number effectively<00:02:01.520> and<00:02:01.840> in<00:02:02.039> this<00:02:02.280> context<00:02:03.280> a<00:02:04.119> multi 00:02:04.510 --> 00:02:04.520 align:start position:0% effectively and in this context a multi 00:02:04.520 --> 00:02:07.350 align:start position:0% effectively and in this context a multi set<00:02:04.719> check<00:02:05.479> would<00:02:05.719> check<00:02:06.079> that<00:02:06.320> the<00:02:06.600> multi<00:02:07.039> set 00:02:07.350 --> 00:02:07.360 align:start position:0% set check would check that the multi set 00:02:07.360 --> 00:02:10.469 align:start position:0% set check would check that the multi set represented<00:02:08.080> by<00:02:08.239> the<00:02:08.399> green<00:02:09.280> squares<00:02:10.239> is 00:02:10.469 --> 00:02:10.479 align:start position:0% represented by the green squares is 00:02:10.479 --> 00:02:12.150 align:start position:0% represented by the green squares is equal<00:02:10.759> to<00:02:10.920> the<00:02:11.000> multi<00:02:11.360> set<00:02:11.560> represented<00:02:12.040> by 00:02:12.150 --> 00:02:12.160 align:start position:0% equal to the multi set represented by 00:02:12.160 --> 00:02:14.190 align:start position:0% equal to the multi set represented by the<00:02:12.280> blue<00:02:12.520> ones<00:02:13.239> so<00:02:13.480> they're<00:02:13.959> they're 00:02:14.190 --> 00:02:14.200 align:start position:0% the blue ones so they're they're 00:02:14.200 --> 00:02:17.509 align:start position:0% the blue ones so they're they're basically<00:02:14.599> multi<00:02:14.959> sets<00:02:15.160> on<00:02:15.400> values<00:02:15.720> in<00:02:15.920> your 00:02:17.509 --> 00:02:17.519 align:start position:0% basically multi sets on values in your 00:02:17.519 --> 00:02:19.589 align:start position:0% basically multi sets on values in your Trace<00:02:18.519> multi<00:02:18.800> set 00:02:19.589 --> 00:02:19.599 align:start position:0% Trace multi set 00:02:19.599 --> 00:02:22.030 align:start position:0% Trace multi set checks<00:02:20.599> um<00:02:21.040> so<00:02:21.200> we'll<00:02:21.400> jump<00:02:21.599> straight<00:02:21.840> into 00:02:22.030 --> 00:02:22.040 align:start position:0% checks um so we'll jump straight into 00:02:22.040 --> 00:02:23.750 align:start position:0% checks um so we'll jump straight into the<00:02:22.160> equation<00:02:22.800> unfortunately<00:02:23.319> we<00:02:23.440> don't<00:02:23.599> have 00:02:23.750 --> 00:02:23.760 align:start position:0% the equation unfortunately we don't have 00:02:23.760 --> 00:02:26.070 align:start position:0% the equation unfortunately we don't have time<00:02:23.959> to<00:02:24.280> talk<00:02:24.480> about<00:02:24.760> why<00:02:24.920> that<00:02:25.040> makes<00:02:25.319> sense 00:02:26.070 --> 00:02:26.080 align:start position:0% time to talk about why that makes sense 00:02:26.080 --> 00:02:28.790 align:start position:0% time to talk about why that makes sense uh<00:02:26.200> it's<00:02:26.319> in<00:02:26.440> the<00:02:26.599> book<00:02:27.440> but<00:02:27.599> we'll<00:02:28.400> take<00:02:28.560> some 00:02:28.790 --> 00:02:28.800 align:start position:0% uh it's in the book but we'll take some 00:02:28.800 --> 00:02:32.509 align:start position:0% uh it's in the book but we'll take some time<00:02:29.080> and<00:02:29.319> and<00:02:29.480> analyze<00:02:30.040> it<00:02:30.360> together<00:02:31.360> so<00:02:32.319> uh 00:02:32.509 --> 00:02:32.519 align:start position:0% time and and analyze it together so uh 00:02:32.519 --> 00:02:35.509 align:start position:0% time and and analyze it together so uh log<00:02:32.760> up<00:02:33.160> is<00:02:33.599> there's<00:02:34.080> four<00:02:34.440> sets<00:02:34.680> of<00:02:34.840> variables 00:02:35.509 --> 00:02:35.519 align:start position:0% log up is there's four sets of variables 00:02:35.519 --> 00:02:40.949 align:start position:0% log up is there's four sets of variables of<00:02:36.000> interest<00:02:37.000> um<00:02:37.640> the<00:02:38.120> a<00:02:38.519> AIG<00:02:38.879> J<00:02:39.120> and<00:02:39.280> B<00:02:39.680> J<00:02:40.680> uh 00:02:40.949 --> 00:02:40.959 align:start position:0% of interest um the a AIG J and B J uh 00:02:40.959 --> 00:02:43.589 align:start position:0% of interest um the a AIG J and B J uh are<00:02:41.239> the<00:02:41.400> elements<00:02:41.879> in<00:02:42.040> your<00:02:42.560> multi<00:02:43.000> set<00:02:43.200> A<00:02:43.440> and 00:02:43.589 --> 00:02:43.599 align:start position:0% are the elements in your multi set A and 00:02:43.599 --> 00:02:47.670 align:start position:0% are the elements in your multi set A and B<00:02:43.879> correspond<00:02:44.560> uh<00:02:45.200> respectively<00:02:46.200> and<00:02:46.480> then<00:02:47.200> uh 00:02:47.670 --> 00:02:47.680 align:start position:0% B correspond uh respectively and then uh 00:02:47.680 --> 00:02:50.830 align:start position:0% B correspond uh respectively and then uh the<00:02:47.879> m<00:02:48.280> a<00:02:48.560> and<00:02:48.720> the<00:02:48.840> MB<00:02:49.319> are<00:02:49.519> the<00:02:49.840> corresponding 00:02:50.830 --> 00:02:50.840 align:start position:0% the m a and the MB are the corresponding 00:02:50.840 --> 00:02:53.470 align:start position:0% the m a and the MB are the corresponding multiplicities<00:02:51.599> of<00:02:51.760> those<00:02:52.000> elements<00:02:53.000> so<00:02:53.319> for 00:02:53.470 --> 00:02:53.480 align:start position:0% multiplicities of those elements so for 00:02:53.480 --> 00:02:55.550 align:start position:0% multiplicities of those elements so for example<00:02:53.920> if<00:02:54.040> you<00:02:54.120> want<00:02:54.319> an<00:02:54.480> element<00:02:54.879> to<00:02:55.080> appear 00:02:55.550 --> 00:02:55.560 align:start position:0% example if you want an element to appear 00:02:55.560 --> 00:02:57.110 align:start position:0% example if you want an element to appear five<00:02:55.800> times<00:02:56.040> in<00:02:56.159> your<00:02:56.280> multiset<00:02:56.920> but<00:02:57.000> it's 00:02:57.110 --> 00:02:57.120 align:start position:0% five times in your multiset but it's 00:02:57.120 --> 00:02:58.710 align:start position:0% five times in your multiset but it's only<00:02:57.360> once<00:02:57.599> in<00:02:57.720> your<00:02:57.840> Trace<00:02:58.200> you<00:02:58.319> could<00:02:58.480> set 00:02:58.710 --> 00:02:58.720 align:start position:0% only once in your Trace you could set 00:02:58.720 --> 00:03:00.390 align:start position:0% only once in your Trace you could set that<00:02:58.879> M<00:02:59.080> to<00:02:59.239> be<00:02:59.480> five 00:03:00.390 --> 00:03:00.400 align:start position:0% that M to be five 00:03:00.400 --> 00:03:02.949 align:start position:0% that M to be five and<00:03:00.599> it's<00:03:00.720> going<00:03:00.840> to<00:03:00.959> be<00:03:01.200> the<00:03:01.480> exact<00:03:01.800> same 00:03:02.949 --> 00:03:02.959 align:start position:0% and it's going to be the exact same 00:03:02.959 --> 00:03:05.750 align:start position:0% and it's going to be the exact same effect<00:03:03.959> so 00:03:05.750 --> 00:03:05.760 align:start position:0% effect so 00:03:05.760 --> 00:03:09.670 align:start position:0% effect so a<00:03:06.760> multiset<00:03:07.640> uh<00:03:08.319> element<00:03:08.879> is<00:03:09.400> the<00:03:09.560> the 00:03:09.670 --> 00:03:09.680 align:start position:0% a multiset uh element is the the 00:03:09.680 --> 00:03:11.630 align:start position:0% a multiset uh element is the the expression<00:03:10.080> is<00:03:10.280> really<00:03:10.599> its<00:03:10.799> multiplicity 00:03:11.630 --> 00:03:11.640 align:start position:0% expression is really its multiplicity 00:03:11.640 --> 00:03:15.470 align:start position:0% expression is really its multiplicity over<00:03:12.280> Alpha<00:03:12.720> a<00:03:12.879> random<00:03:13.239> value<00:03:14.080> minus<00:03:14.640> AIG<00:03:15.319> the 00:03:15.470 --> 00:03:15.480 align:start position:0% over Alpha a random value minus AIG the 00:03:15.480 --> 00:03:17.030 align:start position:0% over Alpha a random value minus AIG the the<00:03:15.599> element 00:03:17.030 --> 00:03:17.040 align:start position:0% the element 00:03:17.040 --> 00:03:21.589 align:start position:0% the element AIG<00:03:18.040> so<00:03:18.599> we<00:03:18.760> have<00:03:19.040> these<00:03:19.720> two<00:03:20.519> uh<00:03:21.159> these<00:03:21.360> two 00:03:21.589 --> 00:03:21.599 align:start position:0% AIG so we have these two uh these two 00:03:21.599 --> 00:03:25.630 align:start position:0% AIG so we have these two uh these two terms<00:03:22.000> per<00:03:22.640> or<00:03:23.000> actually<00:03:23.280> K<00:03:23.480> of<00:03:23.760> them<00:03:24.760> per<00:03:25.080> row 00:03:25.630 --> 00:03:25.640 align:start position:0% terms per or actually K of them per row 00:03:25.640 --> 00:03:28.270 align:start position:0% terms per or actually K of them per row where<00:03:25.920> the<00:03:26.200> I<00:03:26.640> for<00:03:26.840> the<00:03:27.000> row<00:03:27.280> goes<00:03:27.560> from<00:03:27.760> 1<00:03:27.959> to<00:03:28.120> n 00:03:28.270 --> 00:03:28.280 align:start position:0% where the I for the row goes from 1 to n 00:03:28.280 --> 00:03:31.229 align:start position:0% where the I for the row goes from 1 to n minus<00:03:28.560> one<00:03:29.439> the<00:03:29.560> big<00:03:29.840> take<00:03:30.040> away<00:03:30.200> from<00:03:30.400> this<00:03:30.720> is 00:03:31.229 --> 00:03:31.239 align:start position:0% minus one the big take away from this is 00:03:31.239 --> 00:03:34.110 align:start position:0% minus one the big take away from this is log<00:03:31.519> up<00:03:31.799> is<00:03:32.120> one<00:03:32.400> big<00:03:32.640> sum<00:03:33.040> of<00:03:33.239> fractions 00:03:34.110 --> 00:03:34.120 align:start position:0% log up is one big sum of fractions 00:03:34.120 --> 00:03:35.630 align:start position:0% log up is one big sum of fractions that's<00:03:34.360> the<00:03:34.560> one<00:03:34.760> main<00:03:35.000> thing<00:03:35.200> to<00:03:35.319> get<00:03:35.480> from 00:03:35.630 --> 00:03:35.640 align:start position:0% that's the one main thing to get from 00:03:35.640 --> 00:03:37.309 align:start position:0% that's the one main thing to get from this<00:03:35.799> slide<00:03:36.680> and<00:03:36.760> we're<00:03:36.920> going<00:03:37.000> to<00:03:37.080> have<00:03:37.200> to 00:03:37.309 --> 00:03:37.319 align:start position:0% this slide and we're going to have to 00:03:37.319 --> 00:03:38.910 align:start position:0% this slide and we're going to have to prove 00:03:38.910 --> 00:03:38.920 align:start position:0% prove 00:03:38.920 --> 00:03:42.470 align:start position:0% prove this<00:03:39.920> so<00:03:40.080> how<00:03:40.159> do<00:03:40.280> we<00:03:40.360> do<00:03:40.480> it<00:03:40.599> with 00:03:42.470 --> 00:03:42.480 align:start position:0% this so how do we do it with 00:03:42.480 --> 00:03:46.149 align:start position:0% this so how do we do it with Starks<00:03:43.480> um<00:03:43.879> so<00:03:44.040> that's<00:03:44.200> our<00:03:44.360> equation<00:03:44.959> at<00:03:45.080> the 00:03:46.149 --> 00:03:46.159 align:start position:0% Starks um so that's our equation at the 00:03:46.159 --> 00:03:49.630 align:start position:0% Starks um so that's our equation at the bottom<00:03:47.159> and<00:03:47.680> we'll<00:03:47.879> just<00:03:48.000> use<00:03:48.200> an<00:03:48.400> example<00:03:48.959> so 00:03:49.630 --> 00:03:49.640 align:start position:0% bottom and we'll just use an example so 00:03:49.640 --> 00:03:52.710 align:start position:0% bottom and we'll just use an example so we<00:03:49.760> have<00:03:49.879> a<00:03:50.040> main<00:03:50.280> trace<00:03:50.799> of<00:03:51.159> two<00:03:51.560> columns<00:03:52.560> A 00:03:52.710 --> 00:03:52.720 align:start position:0% we have a main trace of two columns A 00:03:52.720 --> 00:03:56.309 align:start position:0% we have a main trace of two columns A and<00:03:52.920> B<00:03:53.640> and<00:03:54.159> each<00:03:54.400> column<00:03:54.840> contains<00:03:55.439> our<00:03:56.200> one 00:03:56.309 --> 00:03:56.319 align:start position:0% and B and each column contains our one 00:03:56.319 --> 00:03:58.550 align:start position:0% and B and each column contains our one of<00:03:56.439> the<00:03:56.560> two<00:03:56.720> multi<00:03:57.120> sets<00:03:57.640> so<00:03:57.879> column<00:03:58.200> A 00:03:58.550 --> 00:03:58.560 align:start position:0% of the two multi sets so column A 00:03:58.560 --> 00:04:01.030 align:start position:0% of the two multi sets so column A contains<00:03:58.879> multi<00:03:59.239> set<00:03:59.519> a<00:04:00.000> and<00:04:00.439> column<00:04:00.760> B 00:04:01.030 --> 00:04:01.040 align:start position:0% contains multi set a and column B 00:04:01.040 --> 00:04:03.830 align:start position:0% contains multi set a and column B contains<00:04:01.519> multi<00:04:01.920> set<00:04:02.120> B<00:04:02.879> and<00:04:03.079> we<00:04:03.200> can<00:04:03.400> visually 00:04:03.830 --> 00:04:03.840 align:start position:0% contains multi set B and we can visually 00:04:03.840 --> 00:04:05.949 align:start position:0% contains multi set B and we can visually inspect<00:04:04.239> that<00:04:04.400> the<00:04:04.519> two<00:04:04.680> multisets<00:04:05.239> are<00:04:05.400> equal 00:04:05.949 --> 00:04:05.959 align:start position:0% inspect that the two multisets are equal 00:04:05.959 --> 00:04:10.949 align:start position:0% inspect that the two multisets are equal they're<00:04:06.360> just<00:04:06.480> a<00:04:06.640> permutation<00:04:07.200> of<00:04:07.480> one<00:04:08.280> to<00:04:08.480> the 00:04:10.949 --> 00:04:10.959 align:start position:0% 00:04:10.959 --> 00:04:14.270 align:start position:0% next<00:04:11.959> so<00:04:12.280> the<00:04:12.959> algorithm<00:04:13.480> to<00:04:13.720> actually<00:04:14.000> do 00:04:14.270 --> 00:04:14.280 align:start position:0% next so the algorithm to actually do 00:04:14.280 --> 00:04:18.150 align:start position:0% next so the algorithm to actually do that<00:04:14.879> to<00:04:15.200> implement<00:04:15.640> the<00:04:16.000> log<00:04:16.280> up<00:04:17.079> is<00:04:17.320> to<00:04:17.639> add 00:04:18.150 --> 00:04:18.160 align:start position:0% that to implement the log up is to add 00:04:18.160 --> 00:04:20.670 align:start position:0% that to implement the log up is to add an<00:04:18.320> extra<00:04:18.639> column<00:04:19.040> that<00:04:19.120> we'll<00:04:19.320> call<00:04:19.600> P<00:04:20.400> and<00:04:20.560> we 00:04:20.670 --> 00:04:20.680 align:start position:0% an extra column that we'll call P and we 00:04:20.680 --> 00:04:24.590 align:start position:0% an extra column that we'll call P and we say<00:04:20.840> it's<00:04:20.919> a<00:04:21.079> running<00:04:21.359> sum<00:04:22.120> column<00:04:23.120> uh<00:04:23.800> so<00:04:24.479> how 00:04:24.590 --> 00:04:24.600 align:start position:0% say it's a running sum column uh so how 00:04:24.600 --> 00:04:28.030 align:start position:0% say it's a running sum column uh so how is<00:04:24.720> it<00:04:24.880> built<00:04:25.479> it<00:04:25.600> starts<00:04:25.919> at<00:04:26.199> zero<00:04:27.199> and<00:04:27.400> then 00:04:28.030 --> 00:04:28.040 align:start position:0% is it built it starts at zero and then 00:04:28.040 --> 00:04:30.830 align:start position:0% is it built it starts at zero and then following<00:04:28.400> our<00:04:28.720> equation<00:04:29.720> if<00:04:29.840> we<00:04:30.000> look<00:04:30.199> at<00:04:30.400> the 00:04:30.830 --> 00:04:30.840 align:start position:0% following our equation if we look at the 00:04:30.840 --> 00:04:33.629 align:start position:0% following our equation if we look at the first<00:04:31.160> element<00:04:31.720> it's<00:04:32.160> off<00:04:32.440> by<00:04:32.639> one<00:04:33.320> compared 00:04:33.629 --> 00:04:33.639 align:start position:0% first element it's off by one compared 00:04:33.639 --> 00:04:35.749 align:start position:0% first element it's off by one compared to<00:04:33.759> the<00:04:34.039> trace<00:04:35.039> but<00:04:35.280> we<00:04:35.520> all<00:04:35.680> the 00:04:35.749 --> 00:04:35.759 align:start position:0% to the trace but we all the 00:04:35.759 --> 00:04:37.550 align:start position:0% to the trace but we all the multiplicities<00:04:36.479> are<00:04:36.639> one<00:04:37.000> and<00:04:37.080> so<00:04:37.240> we<00:04:37.360> have 00:04:37.550 --> 00:04:37.560 align:start position:0% multiplicities are one and so we have 00:04:37.560 --> 00:04:39.870 align:start position:0% multiplicities are one and so we have one<00:04:37.840> over<00:04:38.080> Alpha<00:04:38.440> minus<00:04:38.759> A1<00:04:39.280> which<00:04:39.400> is<00:04:39.600> the 00:04:39.870 --> 00:04:39.880 align:start position:0% one over Alpha minus A1 which is the 00:04:39.880 --> 00:04:43.150 align:start position:0% one over Alpha minus A1 which is the element<00:04:40.320> in<00:04:40.440> the<00:04:40.600> multi<00:04:41.000> set<00:04:41.160> a<00:04:42.000> minus<00:04:42.600> 1<00:04:42.880> over 00:04:43.150 --> 00:04:43.160 align:start position:0% element in the multi set a minus 1 over 00:04:43.160 --> 00:04:46.070 align:start position:0% element in the multi set a minus 1 over Alpha<00:04:43.479> minus<00:04:43.720> A2<00:04:44.320> the<00:04:44.520> element<00:04:44.840> in<00:04:45.039> B<00:04:45.560> and<00:04:45.800> that 00:04:46.070 --> 00:04:46.080 align:start position:0% Alpha minus A2 the element in B and that 00:04:46.080 --> 00:04:48.469 align:start position:0% Alpha minus A2 the element in B and that again<00:04:46.320> is<00:04:46.720> just<00:04:46.919> from<00:04:47.120> our 00:04:48.469 --> 00:04:48.479 align:start position:0% again is just from our 00:04:48.479 --> 00:04:51.230 align:start position:0% again is just from our equation<00:04:49.479> uh<00:04:50.120> then<00:04:50.360> for<00:04:50.520> the<00:04:50.639> next<00:04:50.840> row<00:04:51.080> we 00:04:51.230 --> 00:04:51.240 align:start position:0% equation uh then for the next row we 00:04:51.240 --> 00:04:53.029 align:start position:0% equation uh then for the next row we initialize<00:04:51.720> it<00:04:51.919> with<00:04:52.120> the<00:04:52.440> the<00:04:52.560> contents<00:04:52.919> of 00:04:53.029 --> 00:04:53.039 align:start position:0% initialize it with the the contents of 00:04:53.039 --> 00:04:55.390 align:start position:0% initialize it with the the contents of the<00:04:53.160> previous<00:04:53.479> row<00:04:54.320> and<00:04:54.759> that<00:04:54.880> means<00:04:55.240> we're 00:04:55.390 --> 00:04:55.400 align:start position:0% the previous row and that means we're 00:04:55.400 --> 00:04:57.029 align:start position:0% the previous row and that means we're basically<00:04:55.720> running<00:04:56.000> an 00:04:57.029 --> 00:04:57.039 align:start position:0% basically running an 00:04:57.039 --> 00:05:00.189 align:start position:0% basically running an accumulator<00:04:58.160> and<00:04:59.160> uh<00:04:59.280> and<00:04:59.400> then<00:04:59.800> add<00:04:59.919> the<00:05:00.039> new 00:05:00.189 --> 00:05:00.199 align:start position:0% accumulator and uh and then add the new 00:05:00.199 --> 00:05:02.909 align:start position:0% accumulator and uh and then add the new elements<00:05:00.639> so<00:05:00.840> 1<00:05:01.120> minus<00:05:01.680> Alpha<00:05:01.960> minus<00:05:02.280> A2<00:05:02.720> is 00:05:02.909 --> 00:05:02.919 align:start position:0% elements so 1 minus Alpha minus A2 is 00:05:02.919 --> 00:05:05.830 align:start position:0% elements so 1 minus Alpha minus A2 is added<00:05:03.320> from<00:05:03.479> the<00:05:03.639> multi<00:05:04.039> set<00:05:04.240> a<00:05:04.560> and<00:05:04.720> we<00:05:04.880> remove 00:05:05.830 --> 00:05:05.840 align:start position:0% added from the multi set a and we remove 00:05:05.840 --> 00:05:10.510 align:start position:0% added from the multi set a and we remove one<00:05:06.520> 1<00:05:06.720> over<00:05:07.000> alpha-<00:05:07.520> A3<00:05:08.120> from<00:05:08.479> from<00:05:08.960> B<00:05:09.960> uh<00:05:10.240> same 00:05:10.510 --> 00:05:10.520 align:start position:0% one 1 over alpha- A3 from from B uh same 00:05:10.520 --> 00:05:13.390 align:start position:0% one 1 over alpha- A3 from from B uh same thing<00:05:11.000> over<00:05:11.320> again<00:05:11.639> we<00:05:12.520> we<00:05:12.800> initialize<00:05:13.280> with 00:05:13.390 --> 00:05:13.400 align:start position:0% thing over again we we initialize with 00:05:13.400 --> 00:05:14.870 align:start position:0% thing over again we we initialize with the<00:05:13.520> previous<00:05:13.840> contents<00:05:14.280> add<00:05:14.400> in<00:05:14.560> the<00:05:14.680> new 00:05:14.870 --> 00:05:14.880 align:start position:0% the previous contents add in the new 00:05:14.880 --> 00:05:17.950 align:start position:0% the previous contents add in the new elements<00:05:15.720> and<00:05:15.880> since<00:05:16.120> this<00:05:16.240> is<00:05:16.400> the<00:05:16.600> last<00:05:16.960> row 00:05:17.950 --> 00:05:17.960 align:start position:0% elements and since this is the last row 00:05:17.960 --> 00:05:19.749 align:start position:0% elements and since this is the last row uh<00:05:18.120> it's<00:05:18.280> going<00:05:18.400> to<00:05:18.840> all<00:05:18.960> the<00:05:19.080> terms<00:05:19.440> cancel 00:05:19.749 --> 00:05:19.759 align:start position:0% uh it's going to all the terms cancel 00:05:19.759 --> 00:05:22.950 align:start position:0% uh it's going to all the terms cancel out<00:05:20.080> and<00:05:20.199> it's<00:05:20.400> equal<00:05:20.639> to<00:05:21.240> zero<00:05:22.240> so<00:05:22.600> the 00:05:22.950 --> 00:05:22.960 align:start position:0% out and it's equal to zero so the 00:05:22.960 --> 00:05:25.990 align:start position:0% out and it's equal to zero so the boundary<00:05:23.479> constraints<00:05:24.479> on<00:05:25.440> on<00:05:25.600> this<00:05:25.759> col 00:05:25.990 --> 00:05:26.000 align:start position:0% boundary constraints on on this col 00:05:26.000 --> 00:05:28.790 align:start position:0% boundary constraints on on this col armor<00:05:26.319> that<00:05:26.440> it<00:05:26.560> must<00:05:26.759> start<00:05:27.000> at<00:05:27.319> zero<00:05:28.319> and<00:05:28.560> end 00:05:28.790 --> 00:05:28.800 align:start position:0% armor that it must start at zero and end 00:05:28.800 --> 00:05:31.469 align:start position:0% armor that it must start at zero and end at<00:05:29.000> zero<00:05:29.400> if<00:05:29.960> your<00:05:30.160> multi<00:05:30.639> if<00:05:30.720> your<00:05:30.919> columns<00:05:31.319> if 00:05:31.469 --> 00:05:31.479 align:start position:0% at zero if your multi if your columns if 00:05:31.479 --> 00:05:33.150 align:start position:0% at zero if your multi if your columns if your<00:05:31.600> multi<00:05:31.919> sets<00:05:32.120> are<00:05:32.319> actually 00:05:33.150 --> 00:05:33.160 align:start position:0% your multi sets are actually 00:05:33.160 --> 00:05:35.390 align:start position:0% your multi sets are actually equal<00:05:34.160> but<00:05:34.319> we<00:05:34.440> also<00:05:34.639> have<00:05:34.759> to<00:05:34.919> check<00:05:35.199> that 00:05:35.390 --> 00:05:35.400 align:start position:0% equal but we also have to check that 00:05:35.400 --> 00:05:38.230 align:start position:0% equal but we also have to check that each<00:05:35.840> row<00:05:36.199> is<00:05:36.800> is<00:05:36.960> built<00:05:37.240> correctly<00:05:38.120> and 00:05:38.230 --> 00:05:38.240 align:start position:0% each row is is built correctly and 00:05:38.240 --> 00:05:39.550 align:start position:0% each row is is built correctly and that's<00:05:38.400> what<00:05:38.479> we<00:05:38.600> do<00:05:38.759> in<00:05:38.840> a<00:05:39.080> transition 00:05:39.550 --> 00:05:39.560 align:start position:0% that's what we do in a transition 00:05:39.560 --> 00:05:41.629 align:start position:0% that's what we do in a transition constraint<00:05:40.240> so<00:05:40.360> we're<00:05:40.479> going<00:05:40.560> to<00:05:40.680> zoom<00:05:40.840> in<00:05:40.960> on 00:05:41.629 --> 00:05:41.639 align:start position:0% constraint so we're going to zoom in on 00:05:41.639 --> 00:05:46.070 align:start position:0% constraint so we're going to zoom in on that<00:05:42.639> uh<00:05:43.160> so<00:05:43.319> we<00:05:43.479> have<00:05:43.680> P<00:05:44.160> here 00:05:46.070 --> 00:05:46.080 align:start position:0% that uh so we have P here 00:05:46.080 --> 00:05:48.990 align:start position:0% that uh so we have P here um<00:05:47.080> and<00:05:47.639> uh 00:05:48.990 --> 00:05:49.000 align:start position:0% um and uh 00:05:49.000 --> 00:05:52.590 align:start position:0% um and uh the<00:05:50.000> the<00:05:50.440> transition<00:05:50.960> constraint<00:05:51.639> is 00:05:52.590 --> 00:05:52.600 align:start position:0% the the transition constraint is 00:05:52.600 --> 00:05:53.830 align:start position:0% the the transition constraint is basically<00:05:53.000> what<00:05:53.120> we<00:05:53.240> just<00:05:53.360> did<00:05:53.520> in<00:05:53.680> the 00:05:53.830 --> 00:05:53.840 align:start position:0% basically what we just did in the 00:05:53.840 --> 00:05:56.990 align:start position:0% basically what we just did in the previous<00:05:54.120> slide<00:05:54.600> so<00:05:54.919> pi+<00:05:55.520> 1<00:05:55.680> so<00:05:55.880> the<00:05:56.039> next<00:05:56.280> row 00:05:56.990 --> 00:05:57.000 align:start position:0% previous slide so pi+ 1 so the next row 00:05:57.000 --> 00:06:00.710 align:start position:0% previous slide so pi+ 1 so the next row is<00:05:57.160> equal<00:05:57.440> to<00:05:57.800> pi<00:05:58.800> plus<00:05:59.680> the<00:05:59.840> new 00:06:00.710 --> 00:06:00.720 align:start position:0% is equal to pi plus the new 00:06:00.720 --> 00:06:04.670 align:start position:0% is equal to pi plus the new terms<00:06:01.720> so<00:06:01.960> when<00:06:02.560> a<00:06:02.759> is<00:06:02.880> equal<00:06:03.120> to<00:06:03.360> one<00:06:04.360> uh<00:06:04.560> yeah 00:06:04.670 --> 00:06:04.680 align:start position:0% terms so when a is equal to one uh yeah 00:06:04.680 --> 00:06:07.670 align:start position:0% terms so when a is equal to one uh yeah only<00:06:04.880> two<00:06:05.280> terms<00:06:06.280> this<00:06:06.400> is<00:06:06.599> not<00:06:07.039> quite<00:06:07.319> right 00:06:07.670 --> 00:06:07.680 align:start position:0% only two terms this is not quite right 00:06:07.680 --> 00:06:11.309 align:start position:0% only two terms this is not quite right because<00:06:08.599> um<00:06:09.599> the<00:06:10.479> constraints<00:06:10.960> must<00:06:11.120> be 00:06:11.309 --> 00:06:11.319 align:start position:0% because um the constraints must be 00:06:11.319 --> 00:06:13.909 align:start position:0% because um the constraints must be polinomial<00:06:12.240> and<00:06:12.520> fractions<00:06:13.039> are<00:06:13.240> not<00:06:13.800> you 00:06:13.909 --> 00:06:13.919 align:start position:0% polinomial and fractions are not you 00:06:13.919 --> 00:06:15.430 align:start position:0% polinomial and fractions are not you can't<00:06:14.080> have<00:06:14.199> a<00:06:14.360> fraction<00:06:14.639> in<00:06:14.759> your<00:06:14.919> polinomial 00:06:15.430 --> 00:06:15.440 align:start position:0% can't have a fraction in your polinomial 00:06:15.440 --> 00:06:17.189 align:start position:0% can't have a fraction in your polinomial so<00:06:15.639> what<00:06:15.759> we<00:06:15.840> need<00:06:15.960> to<00:06:16.120> do<00:06:16.319> is<00:06:16.479> simply<00:06:16.759> multiply 00:06:17.189 --> 00:06:17.199 align:start position:0% so what we need to do is simply multiply 00:06:17.199 --> 00:06:19.070 align:start position:0% so what we need to do is simply multiply out<00:06:17.400> the<00:06:17.560> PO<00:06:18.039> the<00:06:18.280> the 00:06:19.070 --> 00:06:19.080 align:start position:0% out the PO the the 00:06:19.080 --> 00:06:21.309 align:start position:0% out the PO the the denominators<00:06:20.080> and<00:06:20.160> so<00:06:20.360> when<00:06:20.639> K<00:06:20.800> is<00:06:20.880> equal<00:06:21.120> to 00:06:21.309 --> 00:06:21.319 align:start position:0% denominators and so when K is equal to 00:06:21.319 --> 00:06:25.309 align:start position:0% denominators and so when K is equal to one<00:06:21.800> with<00:06:21.960> only<00:06:22.440> two<00:06:22.720> terms<00:06:23.560> uh<00:06:23.960> per<00:06:24.160> row<00:06:25.039> uh<00:06:25.160> we 00:06:25.309 --> 00:06:25.319 align:start position:0% one with only two terms uh per row uh we 00:06:25.319 --> 00:06:27.870 align:start position:0% one with only two terms uh per row uh we get<00:06:25.479> this<00:06:25.759> expression<00:06:26.759> which<00:06:26.919> is<00:06:27.199> a<00:06:27.360> degree<00:06:27.639> 3 00:06:27.870 --> 00:06:27.880 align:start position:0% get this expression which is a degree 3 00:06:27.880 --> 00:06:30.070 align:start position:0% get this expression which is a degree 3 polinomial<00:06:28.479> because<00:06:28.639> we're<00:06:28.840> multiplying 00:06:30.070 --> 00:06:30.080 align:start position:0% polinomial because we're multiplying 00:06:30.080 --> 00:06:31.790 align:start position:0% polinomial because we're multiplying three<00:06:30.280> variables 00:06:31.790 --> 00:06:31.800 align:start position:0% three variables 00:06:31.800 --> 00:06:33.990 align:start position:0% three variables together<00:06:32.800> uh<00:06:32.919> but<00:06:33.039> that<00:06:33.160> was<00:06:33.440> when<00:06:33.680> K<00:06:33.840> was 00:06:33.990 --> 00:06:34.000 align:start position:0% together uh but that was when K was 00:06:34.000 --> 00:06:37.430 align:start position:0% together uh but that was when K was equal<00:06:34.199> to<00:06:34.520> one<00:06:35.520> uh<00:06:35.680> when<00:06:35.880> K<00:06:36.000> is<00:06:36.120> equal<00:06:36.360> to<00:06:36.479> two 00:06:37.430 --> 00:06:37.440 align:start position:0% equal to one uh when K is equal to two 00:06:37.440 --> 00:06:39.670 align:start position:0% equal to one uh when K is equal to two uh<00:06:37.720> we're<00:06:37.880> going<00:06:37.960> to<00:06:38.199> have<00:06:38.720> four<00:06:39.080> terms<00:06:39.520> in 00:06:39.670 --> 00:06:39.680 align:start position:0% uh we're going to have four terms in 00:06:39.680 --> 00:06:41.350 align:start position:0% uh we're going to have four terms in that<00:06:39.880> row<00:06:40.400> so<00:06:40.919> we're<00:06:41.080> going<00:06:41.160> to<00:06:41.240> be 00:06:41.350 --> 00:06:41.360 align:start position:0% that row so we're going to be 00:06:41.360 --> 00:06:43.469 align:start position:0% that row so we're going to be multiplying<00:06:42.120> five<00:06:42.360> elements<00:06:42.720> here<00:06:42.840> so<00:06:43.120> degree 00:06:43.469 --> 00:06:43.479 align:start position:0% multiplying five elements here so degree 00:06:43.479 --> 00:06:46.510 align:start position:0% multiplying five elements here so degree five<00:06:44.199> so<00:06:44.440> as<00:06:44.680> K<00:06:44.880> increases<00:06:45.560> so<00:06:45.880> the<00:06:45.960> more<00:06:46.199> terms 00:06:46.510 --> 00:06:46.520 align:start position:0% five so as K increases so the more terms 00:06:46.520 --> 00:06:49.710 align:start position:0% five so as K increases so the more terms you<00:06:46.880> have<00:06:47.880> in<00:06:48.000> your<00:06:48.120> multi<00:06:48.520> set<00:06:48.680> per<00:06:48.880> Row<00:06:49.599> the 00:06:49.710 --> 00:06:49.720 align:start position:0% you have in your multi set per Row the 00:06:49.720 --> 00:06:52.390 align:start position:0% you have in your multi set per Row the larger<00:06:50.039> the<00:06:50.199> degree<00:06:51.199> the<00:06:51.360> trick<00:06:51.599> is<00:06:51.840> in<00:06:52.000> Starks 00:06:52.390 --> 00:06:52.400 align:start position:0% larger the degree the trick is in Starks 00:06:52.400 --> 00:06:56.749 align:start position:0% larger the degree the trick is in Starks you<00:06:52.560> typically<00:06:53.080> have<00:06:53.400> a<00:06:53.720> bound<00:06:54.280> on<00:06:54.879> your<00:06:55.879> uh<00:06:56.599> on 00:06:56.749 --> 00:06:56.759 align:start position:0% you typically have a bound on your uh on 00:06:56.759 --> 00:06:58.070 align:start position:0% you typically have a bound on your uh on your<00:06:56.960> transition<00:06:57.360> constraints<00:06:57.879> on<00:06:57.960> the 00:06:58.070 --> 00:06:58.080 align:start position:0% your transition constraints on the 00:06:58.080 --> 00:07:00.150 align:start position:0% your transition constraints on the degree<00:06:58.360> of<00:06:58.440> your<00:06:58.520> transition<00:06:58.960> constraints 00:07:00.150 --> 00:07:00.160 align:start position:0% degree of your transition constraints 00:07:00.160 --> 00:07:02.430 align:start position:0% degree of your transition constraints and<00:07:00.319> so<00:07:00.759> how<00:07:00.919> we<00:07:01.080> deal<00:07:01.360> with<00:07:01.560> this<00:07:01.919> is<00:07:02.280> we 00:07:02.430 --> 00:07:02.440 align:start position:0% and so how we deal with this is we 00:07:02.440 --> 00:07:05.589 align:start position:0% and so how we deal with this is we simply<00:07:03.000> make<00:07:03.199> a<00:07:03.360> new<00:07:03.720> column<00:07:04.720> if<00:07:05.000> if<00:07:05.160> only<00:07:05.520> we 00:07:05.589 --> 00:07:05.599 align:start position:0% simply make a new column if if only we 00:07:05.599 --> 00:07:07.430 align:start position:0% simply make a new column if if only we can<00:07:05.759> only<00:07:06.039> have<00:07:06.199> let's<00:07:06.400> say<00:07:06.560> two<00:07:07.039> elements 00:07:07.430 --> 00:07:07.440 align:start position:0% can only have let's say two elements 00:07:07.440 --> 00:07:09.070 align:start position:0% can only have let's say two elements here<00:07:07.639> we'll<00:07:07.800> create<00:07:08.000> a<00:07:08.120> new<00:07:08.319> column<00:07:08.759> again<00:07:08.919> a 00:07:09.070 --> 00:07:09.080 align:start position:0% here we'll create a new column again a 00:07:09.080 --> 00:07:11.550 align:start position:0% here we'll create a new column again a running<00:07:09.360> sum<00:07:09.759> with<00:07:10.199> whatever<00:07:11.039> elements<00:07:11.360> are 00:07:11.550 --> 00:07:11.560 align:start position:0% running sum with whatever elements are 00:07:11.560 --> 00:07:12.950 align:start position:0% running sum with whatever elements are left<00:07:11.720> over<00:07:11.960> that<00:07:12.080> we<00:07:12.160> were<00:07:12.319> not<00:07:12.479> able<00:07:12.680> to<00:07:12.800> do 00:07:12.950 --> 00:07:12.960 align:start position:0% left over that we were not able to do 00:07:12.960 --> 00:07:14.830 align:start position:0% left over that we were not able to do with<00:07:13.039> the<00:07:13.160> previous<00:07:13.479> column<00:07:14.240> and<00:07:14.360> so<00:07:14.479> on<00:07:14.680> and 00:07:14.830 --> 00:07:14.840 align:start position:0% with the previous column and so on and 00:07:14.840 --> 00:07:16.869 align:start position:0% with the previous column and so on and so<00:07:15.039> forth<00:07:15.560> it<00:07:15.639> could<00:07:15.800> go<00:07:15.919> up<00:07:16.039> to 00:07:16.869 --> 00:07:16.879 align:start position:0% so forth it could go up to 00:07:16.879 --> 00:07:19.150 align:start position:0% so forth it could go up to hundreds<00:07:17.879> so<00:07:18.160> again<00:07:18.360> the<00:07:18.479> takeaway<00:07:18.879> of<00:07:19.039> when 00:07:19.150 --> 00:07:19.160 align:start position:0% hundreds so again the takeaway of when 00:07:19.160 --> 00:07:21.670 align:start position:0% hundreds so again the takeaway of when we<00:07:19.319> prove<00:07:19.599> log<00:07:19.840> up<00:07:20.000> with<00:07:20.120> Starks<00:07:21.039> is<00:07:21.280> that<00:07:21.520> we 00:07:21.670 --> 00:07:21.680 align:start position:0% we prove log up with Starks is that we 00:07:21.680 --> 00:07:24.189 align:start position:0% we prove log up with Starks is that we need<00:07:22.160> a<00:07:22.360> collection<00:07:23.080> of<00:07:23.360> running<00:07:23.720> some 00:07:24.189 --> 00:07:24.199 align:start position:0% need a collection of running some 00:07:24.199 --> 00:07:26.990 align:start position:0% need a collection of running some columns<00:07:25.199> main 00:07:26.990 --> 00:07:27.000 align:start position:0% columns main 00:07:27.000 --> 00:07:31.469 align:start position:0% columns main takeaway<00:07:28.000> let's<00:07:28.160> see<00:07:28.319> how<00:07:28.440> we<00:07:28.599> do<00:07:28.720> it<00:07:29.039> with<00:07:29.199> the 00:07:31.469 --> 00:07:31.479 align:start position:0% takeaway let's see how we do it with the 00:07:31.479 --> 00:07:35.469 align:start position:0% takeaway let's see how we do it with the gkr<00:07:32.479> uh<00:07:32.639> gkr<00:07:33.639> uh<00:07:33.879> for<00:07:34.160> context<00:07:34.599> is<00:07:34.759> a<00:07:34.919> protocol 00:07:35.469 --> 00:07:35.479 align:start position:0% gkr uh gkr uh for context is a protocol 00:07:35.479 --> 00:07:38.070 align:start position:0% gkr uh gkr uh for context is a protocol that<00:07:36.160> uh<00:07:36.440> proves<00:07:36.800> the<00:07:36.960> correct<00:07:37.240> evaluation<00:07:37.960> of 00:07:38.070 --> 00:07:38.080 align:start position:0% that uh proves the correct evaluation of 00:07:38.080 --> 00:07:39.189 align:start position:0% that uh proves the correct evaluation of a<00:07:38.240> layered 00:07:39.189 --> 00:07:39.199 align:start position:0% a layered 00:07:39.199 --> 00:07:41.430 align:start position:0% a layered circuit<00:07:40.199> uh<00:07:40.319> so<00:07:40.520> here<00:07:40.639> we<00:07:40.759> have<00:07:40.919> a<00:07:41.039> layered 00:07:41.430 --> 00:07:41.440 align:start position:0% circuit uh so here we have a layered 00:07:41.440 --> 00:07:43.869 align:start position:0% circuit uh so here we have a layered circuit<00:07:42.440> and<00:07:42.919> yeah<00:07:43.080> so<00:07:43.319> effectively<00:07:43.759> the 00:07:43.869 --> 00:07:43.879 align:start position:0% circuit and yeah so effectively the 00:07:43.879 --> 00:07:45.629 align:start position:0% circuit and yeah so effectively the prover<00:07:44.240> evaluates<00:07:44.720> the<00:07:44.840> circuit<00:07:45.360> and<00:07:45.479> then 00:07:45.629 --> 00:07:45.639 align:start position:0% prover evaluates the circuit and then 00:07:45.639 --> 00:07:47.950 align:start position:0% prover evaluates the circuit and then uses<00:07:45.960> gkr<00:07:46.520> to<00:07:46.680> convince<00:07:47.039> the<00:07:47.319> verifier<00:07:47.840> that 00:07:47.950 --> 00:07:47.960 align:start position:0% uses gkr to convince the verifier that 00:07:47.960 --> 00:07:49.469 align:start position:0% uses gkr to convince the verifier that this<00:07:48.080> was<00:07:48.240> done 00:07:49.469 --> 00:07:49.479 align:start position:0% this was done 00:07:49.479 --> 00:07:52.430 align:start position:0% this was done correctly<00:07:50.479> uh<00:07:51.039> with<00:07:51.199> log<00:07:51.479> up<00:07:51.759> we<00:07:51.960> have<00:07:52.199> a 00:07:52.430 --> 00:07:52.440 align:start position:0% correctly uh with log up we have a 00:07:52.440 --> 00:07:56.430 align:start position:0% correctly uh with log up we have a specific<00:07:53.039> circuit<00:07:54.039> that<00:07:54.680> encodes<00:07:55.199> the<00:07:55.360> log<00:07:55.560> up 00:07:56.430 --> 00:07:56.440 align:start position:0% specific circuit that encodes the log up 00:07:56.440 --> 00:07:58.550 align:start position:0% specific circuit that encodes the log up relationship<00:07:57.440> so<00:07:57.840> we're<00:07:58.000> going<00:07:58.120> to<00:07:58.280> spend 00:07:58.550 --> 00:07:58.560 align:start position:0% relationship so we're going to spend 00:07:58.560 --> 00:08:00.110 align:start position:0% relationship so we're going to spend some<00:07:58.720> time<00:07:58.919> thinking<00:07:59.199> about<00:07:59.800> how<00:08:00.000> that 00:08:00.110 --> 00:08:00.120 align:start position:0% some time thinking about how that 00:08:00.120 --> 00:08:02.749 align:start position:0% some time thinking about how that circuit<00:08:00.840> looks<00:08:01.159> like<00:08:01.960> how<00:08:02.080> it's<00:08:02.240> designed 00:08:02.749 --> 00:08:02.759 align:start position:0% circuit looks like how it's designed 00:08:02.759 --> 00:08:07.029 align:start position:0% circuit looks like how it's designed really<00:08:03.840> uh<00:08:04.840> if<00:08:04.960> you<00:08:05.199> recall<00:08:05.599> log<00:08:05.840> up<00:08:06.120> is<00:08:06.319> a<00:08:06.680> sum 00:08:07.029 --> 00:08:07.039 align:start position:0% really uh if you recall log up is a sum 00:08:07.039 --> 00:08:10.670 align:start position:0% really uh if you recall log up is a sum of<00:08:07.639> fractions<00:08:08.639> and<00:08:08.840> so<00:08:09.800> uh<00:08:09.960> over<00:08:10.199> your<00:08:10.360> entire 00:08:10.670 --> 00:08:10.680 align:start position:0% of fractions and so uh over your entire 00:08:10.680 --> 00:08:13.550 align:start position:0% of fractions and so uh over your entire Trace<00:08:11.159> so<00:08:11.400> this<00:08:11.479> is<00:08:11.680> depicted<00:08:12.199> by<00:08:12.440> the<00:08:13.280> the<00:08:13.440> the 00:08:13.550 --> 00:08:13.560 align:start position:0% Trace so this is depicted by the the the 00:08:13.560 --> 00:08:15.830 align:start position:0% Trace so this is depicted by the the the sum<00:08:13.720> of<00:08:13.840> fraction<00:08:14.319> at<00:08:14.440> the<00:08:14.520> bottom<00:08:14.879> here<00:08:15.759> and 00:08:15.830 --> 00:08:15.840 align:start position:0% sum of fraction at the bottom here and 00:08:15.840 --> 00:08:18.869 align:start position:0% sum of fraction at the bottom here and to<00:08:16.039> get<00:08:16.280> the<00:08:16.599> the<00:08:16.759> next<00:08:17.000> layer<00:08:17.599> we're<00:08:17.759> going<00:08:17.879> to 00:08:18.869 --> 00:08:18.879 align:start position:0% to get the the next layer we're going to 00:08:18.879 --> 00:08:25.070 align:start position:0% to get the the next layer we're going to merge<00:08:20.240> uh<00:08:21.240> pairs<00:08:22.240> of<00:08:23.240> uh<00:08:23.479> of<00:08:23.639> these<00:08:23.840> terms<00:08:24.840> by 00:08:25.070 --> 00:08:25.080 align:start position:0% merge uh pairs of uh of these terms by 00:08:25.080 --> 00:08:26.510 align:start position:0% merge uh pairs of uh of these terms by common<00:08:25.360> denominator<00:08:26.000> just<00:08:26.159> like<00:08:26.280> what<00:08:26.400> you 00:08:26.510 --> 00:08:26.520 align:start position:0% common denominator just like what you 00:08:26.520 --> 00:08:29.550 align:start position:0% common denominator just like what you did<00:08:26.639> in<00:08:26.800> high<00:08:27.120> school<00:08:28.120> so<00:08:28.319> on<00:08:28.479> and<00:08:28.639> so<00:08:28.879> forth 00:08:29.550 --> 00:08:29.560 align:start position:0% did in high school so on and so forth 00:08:29.560 --> 00:08:33.990 align:start position:0% did in high school so on and so forth until<00:08:29.800> we<00:08:30.000> get<00:08:30.319> just<00:08:31.080> one<00:08:32.080> uh<00:08:32.519> one<00:08:33.000> fraction 00:08:33.990 --> 00:08:34.000 align:start position:0% until we get just one uh one fraction 00:08:34.000 --> 00:08:35.870 align:start position:0% until we get just one uh one fraction and<00:08:34.760> really<00:08:34.959> we're<00:08:35.080> going<00:08:35.200> to<00:08:35.279> stop<00:08:35.479> at<00:08:35.640> two 00:08:35.870 --> 00:08:35.880 align:start position:0% and really we're going to stop at two 00:08:35.880 --> 00:08:38.630 align:start position:0% and really we're going to stop at two but<00:08:36.000> for<00:08:36.240> for<00:08:36.440> intuition<00:08:36.959> we'll<00:08:37.159> say<00:08:37.519> that<00:08:38.320> we 00:08:38.630 --> 00:08:38.640 align:start position:0% but for for intuition we'll say that we 00:08:38.640 --> 00:08:41.709 align:start position:0% but for for intuition we'll say that we we<00:08:38.800> get<00:08:39.120> just<00:08:39.360> one<00:08:40.080> uh<00:08:40.200> one 00:08:41.709 --> 00:08:41.719 align:start position:0% we get just one uh one 00:08:41.719 --> 00:08:45.030 align:start position:0% we get just one uh one fraction<00:08:43.039> um<00:08:44.039> so<00:08:44.240> that's<00:08:44.440> not<00:08:44.680> really<00:08:44.920> a 00:08:45.030 --> 00:08:45.040 align:start position:0% fraction um so that's not really a 00:08:45.040 --> 00:08:47.470 align:start position:0% fraction um so that's not really a circuit<00:08:45.560> but<00:08:45.720> it's<00:08:46.040> a<00:08:46.399> simple<00:08:46.839> translation<00:08:47.320> to 00:08:47.470 --> 00:08:47.480 align:start position:0% circuit but it's a simple translation to 00:08:47.480 --> 00:08:48.389 align:start position:0% circuit but it's a simple translation to get<00:08:47.600> a 00:08:48.389 --> 00:08:48.399 align:start position:0% get a 00:08:48.399 --> 00:08:51.670 align:start position:0% get a circuit<00:08:49.399> uh<00:08:49.920> basically<00:08:50.920> every<00:08:51.160> note<00:08:51.440> in<00:08:51.560> your 00:08:51.670 --> 00:08:51.680 align:start position:0% circuit uh basically every note in your 00:08:51.680 --> 00:08:53.150 align:start position:0% circuit uh basically every note in your circuit<00:08:52.000> is<00:08:52.120> going<00:08:52.200> to<00:08:52.320> have<00:08:52.440> two<00:08:52.640> numbers 00:08:53.150 --> 00:08:53.160 align:start position:0% circuit is going to have two numbers 00:08:53.160 --> 00:08:55.230 align:start position:0% circuit is going to have two numbers corresponding<00:08:53.720> to<00:08:53.839> the<00:08:53.959> numerator<00:08:54.399> and<00:08:54.560> the 00:08:55.230 --> 00:08:55.240 align:start position:0% corresponding to the numerator and the 00:08:55.240 --> 00:08:58.110 align:start position:0% corresponding to the numerator and the denominator<00:08:56.240> and<00:08:56.360> then<00:08:56.560> to<00:08:57.040> merge<00:08:57.440> two<00:08:57.640> nodes 00:08:58.110 --> 00:08:58.120 align:start position:0% denominator and then to merge two nodes 00:08:58.120 --> 00:09:01.389 align:start position:0% denominator and then to merge two nodes together<00:08:58.600> we<00:08:59.560> we<00:08:59.760> apply<00:09:00.240> the<00:09:00.560> the<00:09:00.959> the<00:09:01.240> which 00:09:01.389 --> 00:09:01.399 align:start position:0% together we we apply the the the which 00:09:01.399 --> 00:09:03.949 align:start position:0% together we we apply the the the which we<00:09:01.680> Define<00:09:02.000> as<00:09:02.120> the<00:09:02.240> sum<00:09:02.480> over<00:09:02.680> those<00:09:02.880> pairs<00:09:03.720> is 00:09:03.949 --> 00:09:03.959 align:start position:0% we Define as the sum over those pairs is 00:09:03.959 --> 00:09:05.949 align:start position:0% we Define as the sum over those pairs is really<00:09:04.320> the<00:09:04.800> sum<00:09:05.040> of<00:09:05.200> fractions<00:09:05.640> so<00:09:05.800> the 00:09:05.949 --> 00:09:05.959 align:start position:0% really the sum of fractions so the 00:09:05.959 --> 00:09:08.470 align:start position:0% really the sum of fractions so the denominator<00:09:06.600> here<00:09:06.760> if<00:09:06.880> we<00:09:07.000> look<00:09:07.160> at<00:09:07.959> uh<00:09:08.079> ad<00:09:08.279> D 00:09:08.470 --> 00:09:08.480 align:start position:0% denominator here if we look at uh ad D 00:09:08.480 --> 00:09:10.710 align:start position:0% denominator here if we look at uh ad D plus<00:09:08.680> CB<00:09:09.120> that's<00:09:09.279> the<00:09:09.480> denominator<00:09:10.240> that<00:09:10.519> we 00:09:10.710 --> 00:09:10.720 align:start position:0% plus CB that's the denominator that we 00:09:10.720 --> 00:09:13.269 align:start position:0% plus CB that's the denominator that we got<00:09:10.959> from<00:09:11.399> those<00:09:11.600> two<00:09:11.800> fractions<00:09:12.680> and<00:09:12.760> then<00:09:12.920> BD 00:09:13.269 --> 00:09:13.279 align:start position:0% got from those two fractions and then BD 00:09:13.279 --> 00:09:15.389 align:start position:0% got from those two fractions and then BD is<00:09:13.480> the<00:09:13.959> sorry<00:09:14.320> that<00:09:14.399> was<00:09:14.519> the<00:09:14.640> numerator<00:09:15.160> and 00:09:15.389 --> 00:09:15.399 align:start position:0% is the sorry that was the numerator and 00:09:15.399 --> 00:09:18.470 align:start position:0% is the sorry that was the numerator and BD<00:09:15.640> is<00:09:15.720> the 00:09:18.470 --> 00:09:18.480 align:start position:0% 00:09:18.480 --> 00:09:21.110 align:start position:0% denominator<00:09:19.480> so<00:09:19.720> then<00:09:20.000> what<00:09:20.079> do<00:09:20.240> we<00:09:20.360> do<00:09:20.600> with 00:09:21.110 --> 00:09:21.120 align:start position:0% denominator so then what do we do with 00:09:21.120 --> 00:09:23.790 align:start position:0% denominator so then what do we do with uh<00:09:21.720> G<00:09:22.160> on<00:09:22.240> this<00:09:22.399> circuit<00:09:23.000> there's<00:09:23.560> actually 00:09:23.790 --> 00:09:23.800 align:start position:0% uh G on this circuit there's actually 00:09:23.800 --> 00:09:25.790 align:start position:0% uh G on this circuit there's actually slight<00:09:24.079> modification<00:09:24.600> to<00:09:24.760> the<00:09:24.839> gkr<00:09:25.360> protocol 00:09:25.790 --> 00:09:25.800 align:start position:0% slight modification to the gkr protocol 00:09:25.800 --> 00:09:26.949 align:start position:0% slight modification to the gkr protocol but<00:09:26.079> we're<00:09:26.279> we're<00:09:26.360> not<00:09:26.480> going<00:09:26.600> to<00:09:26.680> have<00:09:26.839> time 00:09:26.949 --> 00:09:26.959 align:start position:0% but we're we're not going to have time 00:09:26.959 --> 00:09:28.949 align:start position:0% but we're we're not going to have time to<00:09:27.120> go<00:09:27.279> through<00:09:27.560> this<00:09:28.279> again<00:09:28.560> that's<00:09:28.720> all<00:09:28.839> in 00:09:28.949 --> 00:09:28.959 align:start position:0% to go through this again that's all in 00:09:28.959 --> 00:09:30.069 align:start position:0% to go through this again that's all in the<00:09:29.079> book 00:09:30.069 --> 00:09:30.079 align:start position:0% the book 00:09:30.079 --> 00:09:32.509 align:start position:0% the book uh<00:09:30.200> so<00:09:30.360> at<00:09:30.440> a<00:09:30.600> high<00:09:30.839> level<00:09:31.399> theover<00:09:32.040> evaluates 00:09:32.509 --> 00:09:32.519 align:start position:0% uh so at a high level theover evaluates 00:09:32.519 --> 00:09:33.470 align:start position:0% uh so at a high level theover evaluates this 00:09:33.470 --> 00:09:33.480 align:start position:0% this 00:09:33.480 --> 00:09:36.590 align:start position:0% this circuit<00:09:34.480> uh<00:09:34.680> and<00:09:34.839> then<00:09:35.760> sends<00:09:36.160> the<00:09:36.279> output 00:09:36.590 --> 00:09:36.600 align:start position:0% circuit uh and then sends the output 00:09:36.600 --> 00:09:39.150 align:start position:0% circuit uh and then sends the output layer<00:09:36.800> to<00:09:36.920> the<00:09:37.000> verifier<00:09:37.880> so<00:09:38.279> only<00:09:38.519> two 00:09:39.150 --> 00:09:39.160 align:start position:0% layer to the verifier so only two 00:09:39.160 --> 00:09:42.230 align:start position:0% layer to the verifier so only two fractions<00:09:40.160> that<00:09:40.399> represent<00:09:40.880> the<00:09:41.040> exact<00:09:41.360> same 00:09:42.230 --> 00:09:42.240 align:start position:0% fractions that represent the exact same 00:09:42.240 --> 00:09:44.190 align:start position:0% fractions that represent the exact same uh<00:09:42.480> exact<00:09:42.760> same<00:09:43.000> sum<00:09:43.440> that<00:09:43.519> was<00:09:43.640> in<00:09:43.760> the<00:09:43.880> input 00:09:44.190 --> 00:09:44.200 align:start position:0% uh exact same sum that was in the input 00:09:44.200 --> 00:09:48.069 align:start position:0% uh exact same sum that was in the input layer<00:09:45.000> but<00:09:45.800> really<00:09:46.079> at<00:09:46.640> on<00:09:47.360> with<00:09:47.560> two 00:09:48.069 --> 00:09:48.079 align:start position:0% layer but really at on with two 00:09:48.079 --> 00:09:51.150 align:start position:0% layer but really at on with two fractions<00:09:49.079> so<00:09:49.320> the<00:09:49.600> verifier<00:09:50.079> can<00:09:50.279> easily 00:09:51.150 --> 00:09:51.160 align:start position:0% fractions so the verifier can easily 00:09:51.160 --> 00:09:54.269 align:start position:0% fractions so the verifier can easily check<00:09:52.160> uh<00:09:52.279> that<00:09:52.440> this<00:09:52.560> is<00:09:52.720> equal<00:09:52.920> to<00:09:53.160> zero<00:09:54.160> and 00:09:54.269 --> 00:09:54.279 align:start position:0% check uh that this is equal to zero and 00:09:54.279 --> 00:09:56.030 align:start position:0% check uh that this is equal to zero and we'll<00:09:54.519> also<00:09:54.720> check<00:09:54.920> the<00:09:55.120> denominators<00:09:55.640> are 00:09:56.030 --> 00:09:56.040 align:start position:0% we'll also check the denominators are 00:09:56.040 --> 00:09:58.150 align:start position:0% we'll also check the denominators are nonzero<00:09:56.600> to<00:09:56.720> make<00:09:56.839> sure<00:09:57.040> that<00:09:57.160> they're<00:09:57.399> proper 00:09:58.150 --> 00:09:58.160 align:start position:0% nonzero to make sure that they're proper 00:09:58.160 --> 00:10:00.470 align:start position:0% nonzero to make sure that they're proper fractions<00:09:59.160> and<00:09:59.399> and<00:09:59.560> then<00:10:00.000> prover<00:10:00.360> and 00:10:00.470 --> 00:10:00.480 align:start position:0% fractions and and then prover and 00:10:00.480 --> 00:10:02.790 align:start position:0% fractions and and then prover and verifier<00:10:00.959> are<00:10:01.120> going<00:10:01.240> to<00:10:01.720> run<00:10:02.000> gkr<00:10:02.600> to 00:10:02.790 --> 00:10:02.800 align:start position:0% verifier are going to run gkr to 00:10:02.800 --> 00:10:07.949 align:start position:0% verifier are going to run gkr to convince<00:10:03.160> the<00:10:03.680> verifier<00:10:04.680> that<00:10:05.760> uh<00:10:06.800> the<00:10:07.800> the 00:10:07.949 --> 00:10:07.959 align:start position:0% convince the verifier that uh the the 00:10:07.959 --> 00:10:10.150 align:start position:0% convince the verifier that uh the the circuit<00:10:08.480> was<00:10:08.800> properly<00:10:09.240> evaluated<00:10:09.920> and<00:10:10.040> that 00:10:10.150 --> 00:10:10.160 align:start position:0% circuit was properly evaluated and that 00:10:10.160 --> 00:10:13.310 align:start position:0% circuit was properly evaluated and that translates<00:10:10.680> to<00:10:11.640> that<00:10:11.839> the<00:10:12.560> output<00:10:12.959> layer 00:10:13.310 --> 00:10:13.320 align:start position:0% translates to that the output layer 00:10:13.320 --> 00:10:15.990 align:start position:0% translates to that the output layer really<00:10:13.800> comes<00:10:14.120> from<00:10:14.360> the<00:10:14.800> input<00:10:15.160> layer<00:10:15.760> it's 00:10:15.990 --> 00:10:16.000 align:start position:0% really comes from the input layer it's 00:10:16.000 --> 00:10:18.030 align:start position:0% really comes from the input layer it's basically<00:10:16.440> the<00:10:16.560> same<00:10:16.800> sum<00:10:17.200> but<00:10:17.440> represented 00:10:18.030 --> 00:10:18.040 align:start position:0% basically the same sum but represented 00:10:18.040 --> 00:10:20.389 align:start position:0% basically the same sum but represented in<00:10:18.399> a<00:10:18.600> different<00:10:18.959> set<00:10:19.120> of 00:10:20.389 --> 00:10:20.399 align:start position:0% in a different set of 00:10:20.399 --> 00:10:23.949 align:start position:0% in a different set of fractions<00:10:21.399> we're<00:10:21.560> not<00:10:21.760> done<00:10:22.000> yet<00:10:22.959> because 00:10:23.949 --> 00:10:23.959 align:start position:0% fractions we're not done yet because 00:10:23.959 --> 00:10:25.710 align:start position:0% fractions we're not done yet because we've<00:10:24.240> we<00:10:24.399> know<00:10:24.600> that<00:10:24.720> the<00:10:24.839> output<00:10:25.200> layer<00:10:25.440> is 00:10:25.710 --> 00:10:25.720 align:start position:0% we've we know that the output layer is 00:10:25.720 --> 00:10:27.829 align:start position:0% we've we know that the output layer is equivalent<00:10:26.240> to<00:10:26.440> the<00:10:26.600> input<00:10:26.920> layer<00:10:27.560> we<00:10:27.680> don't 00:10:27.829 --> 00:10:27.839 align:start position:0% equivalent to the input layer we don't 00:10:27.839 --> 00:10:29.350 align:start position:0% equivalent to the input layer we don't know<00:10:28.040> yet<00:10:28.160> that<00:10:28.279> the<00:10:28.399> input<00:10:28.680> layer<00:10:28.920> was 00:10:29.350 --> 00:10:29.360 align:start position:0% know yet that the input layer was 00:10:29.360 --> 00:10:33.269 align:start position:0% know yet that the input layer was properly<00:10:29.920> populated<00:10:30.560> from<00:10:31.320> elements<00:10:31.640> of<00:10:31.760> your 00:10:33.269 --> 00:10:33.279 align:start position:0% properly populated from elements of your 00:10:33.279 --> 00:10:37.509 align:start position:0% properly populated from elements of your Trace<00:10:34.360> uh<00:10:35.360> to<00:10:35.560> do<00:10:35.839> that<00:10:36.600> we're<00:10:36.800> going<00:10:36.880> to<00:10:37.399> have 00:10:37.509 --> 00:10:37.519 align:start position:0% Trace uh to do that we're going to have 00:10:37.519 --> 00:10:39.030 align:start position:0% Trace uh to do that we're going to have to<00:10:37.639> do<00:10:37.800> this<00:10:37.920> in<00:10:38.079> Starks<00:10:38.600> and<00:10:38.680> we're<00:10:38.800> going<00:10:38.920> to 00:10:39.030 --> 00:10:39.040 align:start position:0% to do this in Starks and we're going to 00:10:39.040 --> 00:10:41.629 align:start position:0% to do this in Starks and we're going to have<00:10:39.160> to<00:10:39.399> add<00:10:39.639> two<00:10:39.920> columns<00:10:40.440> to<00:10:40.600> our 00:10:41.629 --> 00:10:41.639 align:start position:0% have to add two columns to our 00:10:41.639 --> 00:10:44.310 align:start position:0% have to add two columns to our Trace<00:10:42.639> we<00:10:42.760> will<00:10:42.959> not<00:10:43.160> go<00:10:43.320> into<00:10:43.560> how<00:10:43.720> we<00:10:43.880> do<00:10:44.079> that 00:10:44.310 --> 00:10:44.320 align:start position:0% Trace we will not go into how we do that 00:10:44.320 --> 00:10:46.030 align:start position:0% Trace we will not go into how we do that unfortunately<00:10:45.000> because<00:10:45.240> it's<00:10:45.519> like<00:10:45.760> very 00:10:46.030 --> 00:10:46.040 align:start position:0% unfortunately because it's like very 00:10:46.040 --> 00:10:49.430 align:start position:0% unfortunately because it's like very cool<00:10:46.839> and<00:10:47.240> like<00:10:47.519> quite<00:10:48.399> quite<00:10:48.639> clever 00:10:49.430 --> 00:10:49.440 align:start position:0% cool and like quite quite clever 00:10:49.440 --> 00:10:51.990 align:start position:0% cool and like quite quite clever honestly<00:10:50.440> uh<00:10:50.639> but<00:10:51.000> the<00:10:51.240> the<00:10:51.399> the<00:10:51.560> long<00:10:51.760> story 00:10:51.990 --> 00:10:52.000 align:start position:0% honestly uh but the the the long story 00:10:52.000 --> 00:10:53.550 align:start position:0% honestly uh but the the the long story short<00:10:52.279> is<00:10:52.399> that<00:10:52.519> we<00:10:52.680> need<00:10:52.959> we<00:10:53.120> really<00:10:53.399> only 00:10:53.550 --> 00:10:53.560 align:start position:0% short is that we need we really only 00:10:53.560 --> 00:10:56.350 align:start position:0% short is that we need we really only need<00:10:53.800> two<00:10:54.040> columns<00:10:54.639> in<00:10:55.079> uh<00:10:55.800> in<00:10:55.920> our<00:10:56.120> in<00:10:56.240> our 00:10:56.350 --> 00:10:56.360 align:start position:0% need two columns in uh in our in our 00:10:56.360 --> 00:10:59.590 align:start position:0% need two columns in uh in our in our trees 00:10:59.590 --> 00:10:59.600 align:start position:0% 00:10:59.600 --> 00:11:01.389 align:start position:0% so<00:10:59.800> if<00:10:59.880> we<00:11:00.000> take<00:11:00.160> a<00:11:00.320> step 00:11:01.389 --> 00:11:01.399 align:start position:0% so if we take a step 00:11:01.399 --> 00:11:05.750 align:start position:0% so if we take a step back<00:11:02.399> um<00:11:03.079> when<00:11:03.279> we<00:11:03.399> were<00:11:03.600> proving<00:11:04.040> log<00:11:04.360> up<00:11:04.959> with 00:11:05.750 --> 00:11:05.760 align:start position:0% back um when we were proving log up with 00:11:05.760 --> 00:11:09.710 align:start position:0% back um when we were proving log up with uh<00:11:06.560> with<00:11:06.839> Starks<00:11:07.839> we<00:11:07.959> ended<00:11:08.240> up<00:11:08.440> needing<00:11:09.200> a 00:11:09.710 --> 00:11:09.720 align:start position:0% uh with Starks we ended up needing a 00:11:09.720 --> 00:11:12.509 align:start position:0% uh with Starks we ended up needing a collection<00:11:10.720> of<00:11:10.880> running<00:11:11.160> some 00:11:12.509 --> 00:11:12.519 align:start position:0% collection of running some 00:11:12.519 --> 00:11:16.750 align:start position:0% collection of running some columns<00:11:13.519> again<00:11:13.800> that<00:11:13.920> can<00:11:14.079> go<00:11:14.279> up<00:11:14.399> to<00:11:14.880> 100<00:11:15.519> or 00:11:16.750 --> 00:11:16.760 align:start position:0% columns again that can go up to 100 or 00:11:16.760 --> 00:11:22.190 align:start position:0% columns again that can go up to 100 or more<00:11:17.760> with<00:11:18.440> gkr<00:11:19.440> we<00:11:20.200> needed<00:11:20.560> only<00:11:20.839> two<00:11:21.200> columns 00:11:22.190 --> 00:11:22.200 align:start position:0% more with gkr we needed only two columns 00:11:22.200 --> 00:11:25.110 align:start position:0% more with gkr we needed only two columns but<00:11:22.360> with<00:11:22.519> the<00:11:22.720> overhead<00:11:23.360> of<00:11:24.120> gkr<00:11:24.639> on<00:11:24.839> top<00:11:24.959> of 00:11:25.110 --> 00:11:25.120 align:start position:0% but with the overhead of gkr on top of 00:11:25.120 --> 00:11:27.949 align:start position:0% but with the overhead of gkr on top of that<00:11:26.040> so<00:11:26.200> then<00:11:26.360> the<00:11:26.480> big<00:11:26.720> question<00:11:27.120> that<00:11:27.760> we 00:11:27.949 --> 00:11:27.959 align:start position:0% that so then the big question that we 00:11:27.959 --> 00:11:31.590 align:start position:0% that so then the big question that we asked<00:11:28.360> before<00:11:28.720> starting<00:11:29.440> is<00:11:29.920> which<00:11:30.079> is<00:11:30.600> best 00:11:31.590 --> 00:11:31.600 align:start position:0% asked before starting is which is best 00:11:31.600 --> 00:11:34.670 align:start position:0% asked before starting is which is best and<00:11:32.360> like<00:11:32.800> anything<00:11:33.200> in<00:11:33.399> VMS<00:11:34.160> there's<00:11:34.440> no 00:11:34.670 --> 00:11:34.680 align:start position:0% and like anything in VMS there's no 00:11:34.680 --> 00:11:37.550 align:start position:0% and like anything in VMS there's no simple<00:11:35.240> answer<00:11:36.240> uh<00:11:36.360> it<00:11:36.480> depends<00:11:36.760> on<00:11:36.880> a<00:11:37.079> variety 00:11:37.550 --> 00:11:37.560 align:start position:0% simple answer uh it depends on a variety 00:11:37.560 --> 00:11:40.990 align:start position:0% simple answer uh it depends on a variety of<00:11:38.160> parameters<00:11:39.160> uh<00:11:39.639> so<00:11:39.959> we<00:11:40.120> just<00:11:40.279> went<00:11:40.480> on<00:11:40.720> and 00:11:40.990 --> 00:11:41.000 align:start position:0% of parameters uh so we just went on and 00:11:41.000 --> 00:11:45.190 align:start position:0% of parameters uh so we just went on and and<00:11:41.480> did<00:11:41.639> it<00:11:42.360> and<00:11:42.560> so<00:11:43.519> uh<00:11:43.680> there<00:11:44.480> are<00:11:44.720> takeaways 00:11:45.190 --> 00:11:45.200 align:start position:0% and did it and so uh there are takeaways 00:11:45.200 --> 00:11:47.470 align:start position:0% and did it and so uh there are takeaways from<00:11:45.600> uh<00:11:45.800> from<00:11:45.959> doing 00:11:47.470 --> 00:11:47.480 align:start position:0% from uh from doing 00:11:47.480 --> 00:11:50.310 align:start position:0% from uh from doing this<00:11:48.480> many<00:11:48.760> many<00:11:49.160> many<00:11:49.440> different<00:11:49.760> parameters 00:11:50.310 --> 00:11:50.320 align:start position:0% this many many many different parameters 00:11:50.320 --> 00:11:52.670 align:start position:0% this many many many different parameters influence<00:11:51.279> which<00:11:51.519> technique<00:11:51.880> is<00:11:52.160> best<00:11:52.440> in 00:11:52.670 --> 00:11:52.680 align:start position:0% influence which technique is best in 00:11:52.680 --> 00:11:55.230 align:start position:0% influence which technique is best in what<00:11:52.880> circumstance<00:11:53.720> depending<00:11:54.120> on<00:11:54.920> what<00:11:55.040> you 00:11:55.230 --> 00:11:55.240 align:start position:0% what circumstance depending on what you 00:11:55.240 --> 00:11:58.110 align:start position:0% what circumstance depending on what you care<00:11:55.600> about<00:11:56.600> um<00:11:56.959> we're<00:11:57.120> going<00:11:57.240> to<00:11:57.360> focus<00:11:57.639> on 00:11:58.110 --> 00:11:58.120 align:start position:0% care about um we're going to focus on 00:11:58.120 --> 00:12:01.150 align:start position:0% care about um we're going to focus on three<00:11:59.120> uh<00:11:59.320> hash<00:11:59.560> function<00:11:59.880> speed<00:12:00.639> the<00:12:00.880> degree 00:12:01.150 --> 00:12:01.160 align:start position:0% three uh hash function speed the degree 00:12:01.160 --> 00:12:04.670 align:start position:0% three uh hash function speed the degree of<00:12:01.279> star<00:12:01.720> constraints<00:12:02.720> and<00:12:02.920> the<00:12:03.120> size<00:12:03.320> of<00:12:03.399> the 00:12:04.670 --> 00:12:04.680 align:start position:0% of star constraints and the size of the 00:12:04.680 --> 00:12:08.430 align:start position:0% of star constraints and the size of the bus<00:12:05.680> so<00:12:06.519> hash<00:12:06.800> function<00:12:07.120> speed<00:12:07.839> uh<00:12:08.040> when<00:12:08.320> in 00:12:08.430 --> 00:12:08.440 align:start position:0% bus so hash function speed uh when in 00:12:08.440 --> 00:12:10.389 align:start position:0% bus so hash function speed uh when in Starks<00:12:08.880> you<00:12:09.000> have<00:12:09.120> to<00:12:09.240> choose<00:12:09.480> your<00:12:09.639> hash 00:12:10.389 --> 00:12:10.399 align:start position:0% Starks you have to choose your hash 00:12:10.399 --> 00:12:15.069 align:start position:0% Starks you have to choose your hash function<00:12:11.399> uh<00:12:12.320> and<00:12:13.079> by<00:12:13.839> by<00:12:14.079> slow<00:12:14.440> or<00:12:14.680> fast<00:12:14.920> we 00:12:15.069 --> 00:12:15.079 align:start position:0% function uh and by by slow or fast we 00:12:15.079 --> 00:12:17.670 align:start position:0% function uh and by by slow or fast we really<00:12:15.360> mean<00:12:15.800> on<00:12:16.120> for<00:12:16.320> the<00:12:16.519> prover<00:12:17.040> on<00:12:17.399> on<00:12:17.519> the 00:12:17.670 --> 00:12:17.680 align:start position:0% really mean on for the prover on on the 00:12:17.680 --> 00:12:20.910 align:start position:0% really mean on for the prover on on the PR's<00:12:18.320> machine<00:12:19.320> how<00:12:19.560> fast<00:12:19.800> is<00:12:19.880> it<00:12:20.040> to<00:12:20.399> to<00:12:20.639> Hash 00:12:20.910 --> 00:12:20.920 align:start position:0% PR's machine how fast is it to to Hash 00:12:20.920 --> 00:12:24.110 align:start position:0% PR's machine how fast is it to to Hash two<00:12:21.160> numbers<00:12:21.560> or<00:12:21.760> whatever<00:12:22.120> or<00:12:22.320> how<00:12:22.720> however 00:12:24.110 --> 00:12:24.120 align:start position:0% two numbers or whatever or how however 00:12:24.120 --> 00:12:27.350 align:start position:0% two numbers or whatever or how however many<00:12:25.120> so<00:12:25.560> the<00:12:25.720> reason<00:12:26.199> why<00:12:26.600> the<00:12:26.760> hash<00:12:27.000> function 00:12:27.350 --> 00:12:27.360 align:start position:0% many so the reason why the hash function 00:12:27.360 --> 00:12:30.350 align:start position:0% many so the reason why the hash function speed<00:12:28.160> matters<00:12:29.199> in<00:12:29.440> choosing<00:12:29.880> one<00:12:30.040> or<00:12:30.199> the 00:12:30.350 --> 00:12:30.360 align:start position:0% speed matters in choosing one or the 00:12:30.360 --> 00:12:34.069 align:start position:0% speed matters in choosing one or the other<00:12:30.760> is<00:12:31.800> that<00:12:32.800> if<00:12:32.920> you<00:12:33.120> recall<00:12:33.440> from<00:12:33.600> log<00:12:33.880> up 00:12:34.069 --> 00:12:34.079 align:start position:0% other is that if you recall from log up 00:12:34.079 --> 00:12:36.430 align:start position:0% other is that if you recall from log up we<00:12:34.160> were<00:12:34.399> adding<00:12:34.720> a<00:12:34.839> bunch<00:12:35.000> of<00:12:35.199> columns<00:12:36.079> and 00:12:36.430 --> 00:12:36.440 align:start position:0% we were adding a bunch of columns and 00:12:36.440 --> 00:12:38.870 align:start position:0% we were adding a bunch of columns and each<00:12:36.720> columns<00:12:37.720> adds<00:12:38.040> hashing<00:12:38.399> work<00:12:38.639> to<00:12:38.760> the 00:12:38.870 --> 00:12:38.880 align:start position:0% each columns adds hashing work to the 00:12:38.880 --> 00:12:41.750 align:start position:0% each columns adds hashing work to the prover<00:12:39.600> and<00:12:39.720> so<00:12:40.000> if<00:12:40.360> that<00:12:41.000> hashing<00:12:41.320> work<00:12:41.560> is 00:12:41.750 --> 00:12:41.760 align:start position:0% prover and so if that hashing work is 00:12:41.760 --> 00:12:43.430 align:start position:0% prover and so if that hashing work is expensive<00:12:42.279> because<00:12:42.480> you<00:12:42.600> have<00:12:42.720> a<00:12:42.839> slow<00:12:43.120> hash 00:12:43.430 --> 00:12:43.440 align:start position:0% expensive because you have a slow hash 00:12:43.440 --> 00:12:46.230 align:start position:0% expensive because you have a slow hash function<00:12:44.440> it's<00:12:44.600> going<00:12:44.680> to<00:12:45.440> going<00:12:45.560> to<00:12:45.839> cost<00:12:46.079> a 00:12:46.230 --> 00:12:46.240 align:start position:0% function it's going to going to cost a 00:12:46.240 --> 00:12:47.870 align:start position:0% function it's going to going to cost a lot<00:12:46.440> to<00:12:46.600> add<00:12:46.839> these 00:12:47.870 --> 00:12:47.880 align:start position:0% lot to add these 00:12:47.880 --> 00:12:50.949 align:start position:0% lot to add these columns<00:12:48.880> uh<00:12:49.560> so<00:12:50.000> the<00:12:50.120> green<00:12:50.360> check<00:12:50.639> marks 00:12:50.949 --> 00:12:50.959 align:start position:0% columns uh so the green check marks 00:12:50.959 --> 00:12:54.509 align:start position:0% columns uh so the green check marks means<00:12:51.839> favorable<00:12:52.320> to<00:12:52.440> gkr<00:12:53.079> to<00:12:53.279> to<00:12:53.399> log<00:12:53.560> up 00:12:54.509 --> 00:12:54.519 align:start position:0% means favorable to gkr to to log up 00:12:54.519 --> 00:12:58.430 align:start position:0% means favorable to gkr to to log up gkr<00:12:55.519> and<00:12:55.639> the<00:12:55.760> reason<00:12:56.120> is<00:12:56.920> uh<00:12:57.839> log<00:12:58.079> up<00:12:58.279> in 00:12:58.430 --> 00:12:58.440 align:start position:0% gkr and the reason is uh log up in 00:12:58.440 --> 00:13:01.750 align:start position:0% gkr and the reason is uh log up in Starks<00:12:59.399> needed<00:12:59.720> a<00:12:59.839> bunch<00:13:00.000> of<00:13:00.199> columns<00:13:01.079> in<00:13:01.320> gkr 00:13:01.750 --> 00:13:01.760 align:start position:0% Starks needed a bunch of columns in gkr 00:13:01.760 --> 00:13:04.829 align:start position:0% Starks needed a bunch of columns in gkr we<00:13:01.880> only<00:13:02.120> need<00:13:02.440> needed<00:13:02.760> two<00:13:03.519> and<00:13:03.639> so<00:13:04.399> if<00:13:04.519> we<00:13:04.680> go 00:13:04.829 --> 00:13:04.839 align:start position:0% we only need needed two and so if we go 00:13:04.839 --> 00:13:07.230 align:start position:0% we only need needed two and so if we go from<00:13:05.079> hundreds<00:13:05.560> to<00:13:05.920> two<00:13:06.399> that's<00:13:06.560> a<00:13:06.720> lot<00:13:06.959> less 00:13:07.230 --> 00:13:07.240 align:start position:0% from hundreds to two that's a lot less 00:13:07.240 --> 00:13:08.910 align:start position:0% from hundreds to two that's a lot less work<00:13:07.440> for<00:13:07.600> the<00:13:07.720> prover<00:13:08.360> of<00:13:08.480> course<00:13:08.639> there's 00:13:08.910 --> 00:13:08.920 align:start position:0% work for the prover of course there's 00:13:08.920 --> 00:13:11.670 align:start position:0% work for the prover of course there's overhead<00:13:09.279> from<00:13:09.440> gkr<00:13:10.000> but<00:13:10.160> from<00:13:10.720> if<00:13:10.839> we<00:13:11.279> hone<00:13:11.519> in 00:13:11.670 --> 00:13:11.680 align:start position:0% overhead from gkr but from if we hone in 00:13:11.680 --> 00:13:13.550 align:start position:0% overhead from gkr but from if we hone in on<00:13:11.839> that<00:13:12.000> parameter<00:13:12.480> it's<00:13:12.680> very 00:13:13.550 --> 00:13:13.560 align:start position:0% on that parameter it's very 00:13:13.560 --> 00:13:16.870 align:start position:0% on that parameter it's very good<00:13:14.560> degree<00:13:14.880> of<00:13:14.959> star<00:13:15.279> constraints<00:13:16.160> uh<00:13:16.560> same 00:13:16.870 --> 00:13:16.880 align:start position:0% good degree of star constraints uh same 00:13:16.880 --> 00:13:18.509 align:start position:0% good degree of star constraints uh same idea 00:13:18.509 --> 00:13:18.519 align:start position:0% idea 00:13:18.519 --> 00:13:21.910 align:start position:0% idea so<00:13:19.519> uh<00:13:19.959> if<00:13:20.079> you<00:13:20.240> recall<00:13:20.560> from<00:13:20.760> earlier<00:13:21.279> the 00:13:21.910 --> 00:13:21.920 align:start position:0% so uh if you recall from earlier the 00:13:21.920 --> 00:13:24.310 align:start position:0% so uh if you recall from earlier the degree<00:13:22.240> bound<00:13:22.519> on<00:13:22.600> your<00:13:22.720> star<00:13:23.079> constraints<00:13:24.079> is 00:13:24.310 --> 00:13:24.320 align:start position:0% degree bound on your star constraints is 00:13:24.320 --> 00:13:27.710 align:start position:0% degree bound on your star constraints is what<00:13:25.120> made<00:13:25.480> us<00:13:26.480> is<00:13:26.639> what<00:13:26.800> made<00:13:26.959> us<00:13:27.199> create<00:13:27.519> new 00:13:27.710 --> 00:13:27.720 align:start position:0% what made us is what made us create new 00:13:27.720 --> 00:13:29.790 align:start position:0% what made us is what made us create new columns<00:13:28.240> because<00:13:28.440> we<00:13:28.519> were<00:13:28.680> not<00:13:28.839> able<00:13:29.279> to<00:13:29.399> pack 00:13:29.790 --> 00:13:29.800 align:start position:0% columns because we were not able to pack 00:13:29.800 --> 00:13:32.230 align:start position:0% columns because we were not able to pack more<00:13:30.800> uh<00:13:30.959> we<00:13:31.040> were<00:13:31.199> not<00:13:31.320> able<00:13:31.560> to<00:13:31.720> pack<00:13:31.880> a<00:13:32.000> bunch 00:13:32.230 --> 00:13:32.240 align:start position:0% more uh we were not able to pack a bunch 00:13:32.240 --> 00:13:35.829 align:start position:0% more uh we were not able to pack a bunch of<00:13:33.160> uh<00:13:33.839> fractions<00:13:34.279> in<00:13:34.399> a<00:13:34.519> single<00:13:34.839> column<00:13:35.680> so 00:13:35.829 --> 00:13:35.839 align:start position:0% of uh fractions in a single column so 00:13:35.839 --> 00:13:37.550 align:start position:0% of uh fractions in a single column so the<00:13:36.000> lower<00:13:36.279> the<00:13:36.480> degree<00:13:36.839> the<00:13:36.959> more<00:13:37.240> columns 00:13:37.550 --> 00:13:37.560 align:start position:0% the lower the degree the more columns 00:13:37.560 --> 00:13:40.069 align:start position:0% the lower the degree the more columns you'll<00:13:37.760> need<00:13:38.600> and<00:13:38.720> then<00:13:38.839> the<00:13:39.040> same<00:13:39.240> intuition 00:13:40.069 --> 00:13:40.079 align:start position:0% you'll need and then the same intuition 00:13:40.079 --> 00:13:43.030 align:start position:0% you'll need and then the same intuition comes<00:13:40.519> back<00:13:41.519> uh<00:13:42.000> the<00:13:42.120> lower<00:13:42.440> the<00:13:42.600> degree<00:13:42.959> the 00:13:43.030 --> 00:13:43.040 align:start position:0% comes back uh the lower the degree the 00:13:43.040 --> 00:13:44.710 align:start position:0% comes back uh the lower the degree the more<00:13:43.279> columns<00:13:43.680> the<00:13:43.760> more<00:13:43.959> columns<00:13:44.320> the<00:13:44.399> more 00:13:44.710 --> 00:13:44.720 align:start position:0% more columns the more columns the more 00:13:44.720 --> 00:13:47.590 align:start position:0% more columns the more columns the more the<00:13:44.800> more<00:13:44.959> work<00:13:45.199> for<00:13:45.320> the<00:13:45.680> prover<00:13:46.680> uh<00:13:46.839> and<00:13:46.959> so 00:13:47.590 --> 00:13:47.600 align:start position:0% the more work for the prover uh and so 00:13:47.600 --> 00:13:48.949 align:start position:0% the more work for the prover uh and so removing<00:13:48.040> all<00:13:48.240> these<00:13:48.399> columns<00:13:48.760> when 00:13:48.949 --> 00:13:48.959 align:start position:0% removing all these columns when 00:13:48.959 --> 00:13:51.389 align:start position:0% removing all these columns when switching<00:13:49.279> to<00:13:49.399> gkr<00:13:50.040> is<00:13:50.240> favorable<00:13:50.680> to 00:13:51.389 --> 00:13:51.399 align:start position:0% switching to gkr is favorable to 00:13:51.399 --> 00:13:53.629 align:start position:0% switching to gkr is favorable to gkr<00:13:52.399> again<00:13:52.560> if<00:13:52.680> your<00:13:52.800> degre<00:13:53.160> is<00:13:53.279> high<00:13:53.480> you're 00:13:53.629 --> 00:13:53.639 align:start position:0% gkr again if your degre is high you're 00:13:53.639 --> 00:13:55.509 align:start position:0% gkr again if your degre is high you're removing<00:13:54.079> fewer<00:13:54.480> columns<00:13:54.880> and<00:13:55.000> so<00:13:55.199> maybe<00:13:55.399> the 00:13:55.509 --> 00:13:55.519 align:start position:0% removing fewer columns and so maybe the 00:13:55.519 --> 00:13:56.990 align:start position:0% removing fewer columns and so maybe the gain<00:13:55.759> is<00:13:55.880> not<00:13:56.040> as 00:13:56.990 --> 00:13:57.000 align:start position:0% gain is not as 00:13:57.000 --> 00:13:59.389 align:start position:0% gain is not as good<00:13:58.000> the<00:13:58.160> last<00:13:58.320> parameter<00:13:58.720> is<00:13:58.839> this<00:13:59.160> size<00:13:59.320> of 00:13:59.389 --> 00:13:59.399 align:start position:0% good the last parameter is this size of 00:13:59.399 --> 00:14:01.629 align:start position:0% good the last parameter is this size of the<00:13:59.560> bus<00:14:00.440> uh<00:14:00.560> by<00:14:00.759> size<00:14:00.920> of<00:14:01.000> the<00:14:01.120> bus<00:14:01.320> we<00:14:01.480> really 00:14:01.629 --> 00:14:01.639 align:start position:0% the bus uh by size of the bus we really 00:14:01.639 --> 00:14:03.710 align:start position:0% the bus uh by size of the bus we really mean<00:14:01.920> how<00:14:02.040> many<00:14:02.360> fractions<00:14:02.959> per<00:14:03.160> row<00:14:03.399> are<00:14:03.519> you 00:14:03.710 --> 00:14:03.720 align:start position:0% mean how many fractions per row are you 00:14:03.720 --> 00:14:08.829 align:start position:0% mean how many fractions per row are you adding<00:14:04.560> that<00:14:04.959> K<00:14:05.320> from<00:14:05.600> the<00:14:05.959> previous<00:14:07.279> uh<00:14:08.279> uh<00:14:08.600> K 00:14:08.829 --> 00:14:08.839 align:start position:0% adding that K from the previous uh uh K 00:14:08.839 --> 00:14:10.790 align:start position:0% adding that K from the previous uh uh K from<00:14:08.959> the<00:14:09.160> previous<00:14:09.639> uh 00:14:10.790 --> 00:14:10.800 align:start position:0% from the previous uh 00:14:10.800 --> 00:14:12.550 align:start position:0% from the previous uh equation 00:14:12.550 --> 00:14:12.560 align:start position:0% equation 00:14:12.560 --> 00:14:16.189 align:start position:0% equation so<00:14:13.560> here<00:14:13.839> if<00:14:13.959> you<00:14:14.120> have<00:14:14.240> a<00:14:14.440> very<00:14:14.759> large<00:14:15.160> bus<00:14:16.000> so 00:14:16.189 --> 00:14:16.199 align:start position:0% so here if you have a very large bus so 00:14:16.199 --> 00:14:17.670 align:start position:0% so here if you have a very large bus so meaning<00:14:16.480> you're<00:14:16.759> adding<00:14:17.040> a<00:14:17.199> bunch<00:14:17.440> of 00:14:17.670 --> 00:14:17.680 align:start position:0% meaning you're adding a bunch of 00:14:17.680 --> 00:14:20.310 align:start position:0% meaning you're adding a bunch of fractions<00:14:18.279> per<00:14:18.480> row<00:14:19.040> you're<00:14:19.199> going<00:14:19.279> to<00:14:19.440> need<00:14:20.120> a 00:14:20.310 --> 00:14:20.320 align:start position:0% fractions per row you're going to need a 00:14:20.320 --> 00:14:23.430 align:start position:0% fractions per row you're going to need a lot<00:14:20.720> more<00:14:21.000> columns<00:14:21.839> same<00:14:22.079> intuition<00:14:23.000> so<00:14:23.279> a 00:14:23.430 --> 00:14:23.440 align:start position:0% lot more columns same intuition so a 00:14:23.440 --> 00:14:25.749 align:start position:0% lot more columns same intuition so a large<00:14:23.959> size<00:14:24.480> uh<00:14:24.639> a<00:14:24.800> large<00:14:25.079> bus<00:14:25.440> is<00:14:25.519> going<00:14:25.639> to 00:14:25.749 --> 00:14:25.759 align:start position:0% large size uh a large bus is going to 00:14:25.759 --> 00:14:27.030 align:start position:0% large size uh a large bus is going to mean<00:14:26.040> a<00:14:26.160> lot<00:14:26.279> of<00:14:26.399> columns<00:14:26.800> when<00:14:26.920> you're 00:14:27.030 --> 00:14:27.040 align:start position:0% mean a lot of columns when you're 00:14:27.040 --> 00:14:29.509 align:start position:0% mean a lot of columns when you're proving<00:14:27.320> in<00:14:27.440> in<00:14:27.639> stars<00:14:28.360> and<00:14:28.519> a<00:14:28.639> lot<00:14:28.839> less<00:14:29.279> a<00:14:29.399> lot 00:14:29.509 --> 00:14:29.519 align:start position:0% proving in in stars and a lot less a lot 00:14:29.519 --> 00:14:31.430 align:start position:0% proving in in stars and a lot less a lot of<00:14:29.680> Prov<00:14:30.079> work<00:14:30.320> that<00:14:30.399> you're<00:14:30.600> removing<00:14:31.320> when 00:14:31.430 --> 00:14:31.440 align:start position:0% of Prov work that you're removing when 00:14:31.440 --> 00:14:33.389 align:start position:0% of Prov work that you're removing when you're<00:14:31.560> switching<00:14:31.880> to 00:14:33.389 --> 00:14:33.399 align:start position:0% you're switching to 00:14:33.399 --> 00:14:37.150 align:start position:0% you're switching to gkr<00:14:34.480> so<00:14:35.480> this<00:14:35.600> is<00:14:35.759> our<00:14:36.040> mental<00:14:36.440> model<00:14:36.839> rough 00:14:37.150 --> 00:14:37.160 align:start position:0% gkr so this is our mental model rough 00:14:37.160 --> 00:14:41.189 align:start position:0% gkr so this is our mental model rough meal<00:14:37.440> model<00:14:37.880> for<00:14:38.800> uh<00:14:39.639> when<00:14:40.440> when<00:14:40.560> to<00:14:40.720> use<00:14:41.000> log 00:14:41.189 --> 00:14:41.199 align:start position:0% meal model for uh when when to use log 00:14:41.199 --> 00:14:44.550 align:start position:0% meal model for uh when when to use log up<00:14:41.360> gkr<00:14:41.839> and<00:14:42.000> when<00:14:42.160> not<00:14:42.320> to<00:14:42.480> use<00:14:43.040> it<00:14:44.040> so<00:14:44.320> in<00:14:44.440> the 00:14:44.550 --> 00:14:44.560 align:start position:0% up gkr and when not to use it so in the 00:14:44.560 --> 00:14:49.189 align:start position:0% up gkr and when not to use it so in the case<00:14:44.720> of<00:14:45.279> MVM<00:14:46.600> we<00:14:47.600> the<00:14:47.839> rollup<00:14:48.360> uses<00:14:48.680> a<00:14:48.800> slow 00:14:49.189 --> 00:14:49.199 align:start position:0% case of MVM we the rollup uses a slow 00:14:49.199 --> 00:14:53.069 align:start position:0% case of MVM we the rollup uses a slow hash<00:14:49.839> function<00:14:50.839> uh<00:14:51.040> so<00:14:51.560> that<00:14:51.720> was<00:14:52.560> that<00:14:52.680> was 00:14:53.069 --> 00:14:53.079 align:start position:0% hash function uh so that was that was 00:14:53.079 --> 00:14:55.110 align:start position:0% hash function uh so that was that was good<00:14:53.320> that<00:14:53.399> was<00:14:53.560> quite<00:14:53.759> favorable<00:14:54.759> but<00:14:55.000> the 00:14:55.110 --> 00:14:55.120 align:start position:0% good that was quite favorable but the 00:14:55.120 --> 00:14:57.590 align:start position:0% good that was quite favorable but the Mind<00:14:55.320> VM<00:14:55.600> uses<00:14:56.040> quite<00:14:56.320> high<00:14:57.160> high<00:14:57.279> degree<00:14:57.519> of 00:14:57.590 --> 00:14:57.600 align:start position:0% Mind VM uses quite high high degree of 00:14:57.600 --> 00:14:59.790 align:start position:0% Mind VM uses quite high high degree of start<00:14:57.839> constraints<00:14:58.279> we<00:14:58.399> use<00:14:58.600> nine 00:14:59.790 --> 00:14:59.800 align:start position:0% start constraints we use nine 00:14:59.800 --> 00:15:02.870 align:start position:0% start constraints we use nine uh<00:15:00.120> three<00:15:00.639> is<00:15:01.120> uh<00:15:01.279> is<00:15:01.519> quite<00:15:01.839> popular<00:15:02.440> five<00:15:02.680> is 00:15:02.870 --> 00:15:02.880 align:start position:0% uh three is uh is quite popular five is 00:15:02.880 --> 00:15:05.590 align:start position:0% uh three is uh is quite popular five is also<00:15:03.360> used<00:15:04.120> so<00:15:04.360> we're<00:15:04.600> we're<00:15:04.800> pretty<00:15:05.120> up<00:15:05.399> up 00:15:05.590 --> 00:15:05.600 align:start position:0% also used so we're we're pretty up up 00:15:05.600 --> 00:15:07.990 align:start position:0% also used so we're we're pretty up up there<00:15:05.800> so<00:15:06.000> not<00:15:06.199> very<00:15:06.399> good<00:15:06.519> for<00:15:06.680> log<00:15:06.880> up<00:15:07.040> gkr 00:15:07.990 --> 00:15:08.000 align:start position:0% there so not very good for log up gkr 00:15:08.000 --> 00:15:10.509 align:start position:0% there so not very good for log up gkr also<00:15:08.199> our<00:15:08.360> bus<00:15:08.639> was<00:15:09.160> kind<00:15:09.279> of<00:15:09.440> small<00:15:10.160> we<00:15:10.320> only 00:15:10.509 --> 00:15:10.519 align:start position:0% also our bus was kind of small we only 00:15:10.519 --> 00:15:14.509 align:start position:0% also our bus was kind of small we only had<00:15:10.680> a<00:15:10.800> few<00:15:11.120> extra<00:15:11.399> columns<00:15:11.720> for<00:15:11.880> log<00:15:12.120> up<00:15:13.519> so 00:15:14.509 --> 00:15:14.519 align:start position:0% had a few extra columns for log up so 00:15:14.519 --> 00:15:17.470 align:start position:0% had a few extra columns for log up so basically<00:15:15.040> we<00:15:15.279> by<00:15:15.560> optimizing<00:15:16.440> gkr<00:15:16.959> and<00:15:17.120> gkr 00:15:17.470 --> 00:15:17.480 align:start position:0% basically we by optimizing gkr and gkr 00:15:17.480 --> 00:15:18.990 align:start position:0% basically we by optimizing gkr and gkr is<00:15:17.600> based<00:15:17.800> on<00:15:17.920> some<00:15:18.199> check<00:15:18.440> and<00:15:18.639> there<00:15:18.759> were<00:15:18.920> a 00:15:18.990 --> 00:15:19.000 align:start position:0% is based on some check and there were a 00:15:19.000 --> 00:15:21.430 align:start position:0% is based on some check and there were a lot<00:15:19.279> of<00:15:19.600> optimizations<00:15:20.399> there<00:15:20.920> to<00:15:21.120> get<00:15:21.240> it 00:15:21.430 --> 00:15:21.440 align:start position:0% lot of optimizations there to get it 00:15:21.440 --> 00:15:24.230 align:start position:0% lot of optimizations there to get it down<00:15:21.680> to<00:15:22.160> get<00:15:22.320> log<00:15:22.600> of<00:15:22.800> gkr<00:15:23.320> down<00:15:23.480> to<00:15:23.800> be 00:15:24.230 --> 00:15:24.240 align:start position:0% down to get log of gkr down to be 00:15:24.240 --> 00:15:27.629 align:start position:0% down to get log of gkr down to be equivalent<00:15:24.720> in<00:15:24.920> prover<00:15:25.360> time<00:15:26.040> to<00:15:26.759> uh<00:15:26.920> Starks 00:15:27.629 --> 00:15:27.639 align:start position:0% equivalent in prover time to uh Starks 00:15:27.639 --> 00:15:29.910 align:start position:0% equivalent in prover time to uh Starks only<00:15:28.639> but<00:15:28.759> it's<00:15:28.959> as<00:15:29.000> soon<00:15:29.120> as<00:15:29.199> we<00:15:29.319> went<00:15:29.560> back<00:15:29.720> to 00:15:29.910 --> 00:15:29.920 align:start position:0% only but it's as soon as we went back to 00:15:29.920 --> 00:15:32.629 align:start position:0% only but it's as soon as we went back to a<00:15:30.079> fast<00:15:30.360> hash<00:15:30.560> function<00:15:30.920> we<00:15:31.079> used<00:15:31.319> Blake<00:15:31.600> 3<00:15:32.519> uh 00:15:32.629 --> 00:15:32.639 align:start position:0% a fast hash function we used Blake 3 uh 00:15:32.639 --> 00:15:35.430 align:start position:0% a fast hash function we used Blake 3 uh log<00:15:32.839> up<00:15:32.959> gkr<00:15:33.360> was<00:15:33.480> still<00:15:33.920> slower<00:15:34.920> and<00:15:35.199> uh<00:15:35.319> it 00:15:35.430 --> 00:15:35.440 align:start position:0% log up gkr was still slower and uh it 00:15:35.440 --> 00:15:36.470 align:start position:0% log up gkr was still slower and uh it seemed 00:15:36.470 --> 00:15:36.480 align:start position:0% seemed 00:15:36.480 --> 00:15:39.870 align:start position:0% seemed like<00:15:37.480> maybe<00:15:37.959> it's<00:15:38.800> it<00:15:38.880> would<00:15:39.120> maybe<00:15:39.639> even 00:15:39.870 --> 00:15:39.880 align:start position:0% like maybe it's it would maybe even 00:15:39.880 --> 00:15:43.590 align:start position:0% like maybe it's it would maybe even never<00:15:40.160> reach<00:15:41.000> uh<00:15:41.279> like<00:15:41.480> parody<00:15:42.480> because<00:15:43.199> our 00:15:43.590 --> 00:15:43.600 align:start position:0% never reach uh like parody because our 00:15:43.600 --> 00:15:47.550 align:start position:0% never reach uh like parody because our bus<00:15:43.839> is<00:15:44.000> small<00:15:44.399> and<00:15:44.519> our<00:15:44.680> degree<00:15:45.160> is<00:15:45.839> High 00:15:47.550 --> 00:15:47.560 align:start position:0% bus is small and our degree is High 00:15:47.560 --> 00:15:49.590 align:start position:0% bus is small and our degree is High um<00:15:48.560> we<00:15:48.680> were<00:15:48.880> not<00:15:49.000> the<00:15:49.120> only<00:15:49.279> ones<00:15:49.440> to 00:15:49.590 --> 00:15:49.600 align:start position:0% um we were not the only ones to 00:15:49.600 --> 00:15:52.030 align:start position:0% um we were not the only ones to implement<00:15:50.000> this<00:15:50.199> starkware<00:15:50.800> also<00:15:51.040> did<00:15:51.279> that 00:15:52.030 --> 00:15:52.040 align:start position:0% implement this starkware also did that 00:15:52.040 --> 00:15:54.309 align:start position:0% implement this starkware also did that and<00:15:52.440> um<00:15:52.639> Andrew<00:15:53.000> Milson<00:15:53.440> was<00:15:53.759> uh<00:15:53.920> very<00:15:54.120> nice 00:15:54.309 --> 00:15:54.319 align:start position:0% and um Andrew Milson was uh very nice 00:15:54.319 --> 00:15:56.550 align:start position:0% and um Andrew Milson was uh very nice and<00:15:54.440> shared<00:15:54.800> with<00:15:54.920> us<00:15:55.560> uh<00:15:55.920> the<00:15:56.040> numbers<00:15:56.399> that 00:15:56.550 --> 00:15:56.560 align:start position:0% and shared with us uh the numbers that 00:15:56.560 --> 00:16:01.269 align:start position:0% and shared with us uh the numbers that they<00:15:56.759> got<00:15:57.759> so<00:15:58.759> they<00:15:59.079> they<00:15:59.399> have<00:16:00.199> uh<00:16:00.720> the<00:16:01.120> their 00:16:01.269 --> 00:16:01.279 align:start position:0% they got so they they have uh the their 00:16:01.279 --> 00:16:03.389 align:start position:0% they got so they they have uh the their in<00:16:01.360> their<00:16:01.519> experiments<00:16:02.040> they<00:16:02.160> used<00:16:02.560> Blake<00:16:02.920> 2 00:16:03.389 --> 00:16:03.399 align:start position:0% in their experiments they used Blake 2 00:16:03.399 --> 00:16:06.150 align:start position:0% in their experiments they used Blake 2 as<00:16:03.480> a<00:16:03.639> hash<00:16:03.920> function<00:16:04.279> so<00:16:04.800> quite<00:16:05.120> fast<00:16:05.959> but 00:16:06.150 --> 00:16:06.160 align:start position:0% as a hash function so quite fast but 00:16:06.160 --> 00:16:08.790 align:start position:0% as a hash function so quite fast but they<00:16:06.319> have<00:16:06.639> degree<00:16:07.040> three<00:16:07.279> constraints<00:16:08.279> so 00:16:08.790 --> 00:16:08.800 align:start position:0% they have degree three constraints so 00:16:08.800 --> 00:16:10.749 align:start position:0% they have degree three constraints so very<00:16:09.040> nice<00:16:09.240> for<00:16:09.399> log<00:16:09.600> of<00:16:09.720> gkr<00:16:10.279> and<00:16:10.399> a<00:16:10.560> very 00:16:10.749 --> 00:16:10.759 align:start position:0% very nice for log of gkr and a very 00:16:10.759 --> 00:16:12.910 align:start position:0% very nice for log of gkr and a very large<00:16:11.000> bus<00:16:11.319> like<00:16:11.480> hundreds<00:16:11.759> of<00:16:11.920> columns<00:16:12.800> and 00:16:12.910 --> 00:16:12.920 align:start position:0% large bus like hundreds of columns and 00:16:12.920 --> 00:16:15.790 align:start position:0% large bus like hundreds of columns and they<00:16:13.040> were<00:16:13.199> able<00:16:13.440> to<00:16:13.560> get<00:16:13.720> a<00:16:13.839> 15%<00:16:14.800> Improvement 00:16:15.790 --> 00:16:15.800 align:start position:0% they were able to get a 15% Improvement 00:16:15.800 --> 00:16:17.150 align:start position:0% they were able to get a 15% Improvement uh<00:16:15.959> and<00:16:16.079> they<00:16:16.199> will<00:16:16.440> say<00:16:16.639> that<00:16:16.759> they're<00:16:16.959> still 00:16:17.150 --> 00:16:17.160 align:start position:0% uh and they will say that they're still 00:16:17.160 --> 00:16:18.670 align:start position:0% uh and they will say that they're still not<00:16:17.319> done<00:16:17.560> optimizing<00:16:18.120> so<00:16:18.279> they<00:16:18.399> think<00:16:18.560> they 00:16:18.670 --> 00:16:18.680 align:start position:0% not done optimizing so they think they 00:16:18.680 --> 00:16:20.949 align:start position:0% not done optimizing so they think they can<00:16:18.839> get<00:16:19.000> that<00:16:19.160> even<00:16:19.440> higher<00:16:20.279> but<00:16:20.399> it<00:16:20.560> just 00:16:20.949 --> 00:16:20.959 align:start position:0% can get that even higher but it just 00:16:20.959 --> 00:16:22.790 align:start position:0% can get that even higher but it just goes<00:16:21.120> to<00:16:21.279> show<00:16:21.720> that<00:16:22.000> even<00:16:22.279> though<00:16:22.440> they<00:16:22.600> have 00:16:22.790 --> 00:16:22.800 align:start position:0% goes to show that even though they have 00:16:22.800 --> 00:16:25.670 align:start position:0% goes to show that even though they have a<00:16:23.160> they<00:16:23.279> use<00:16:23.440> a<00:16:23.680> hash<00:16:24.560> hash<00:16:24.800> function<00:16:25.079> that's 00:16:25.670 --> 00:16:25.680 align:start position:0% a they use a hash hash function that's 00:16:25.680 --> 00:16:27.949 align:start position:0% a they use a hash hash function that's fast<00:16:26.319> they<00:16:26.440> were<00:16:26.639> able<00:16:26.880> to<00:16:27.040> make<00:16:27.199> log<00:16:27.440> of<00:16:27.560> gkr 00:16:27.949 --> 00:16:27.959 align:start position:0% fast they were able to make log of gkr 00:16:27.959 --> 00:16:31.350 align:start position:0% fast they were able to make log of gkr worth<00:16:28.160> it<00:16:29.000> again<00:16:29.279> this<00:16:29.360> is<00:16:29.480> all<00:16:29.720> pro<00:16:30.199> time<00:16:31.199> the 00:16:31.350 --> 00:16:31.360 align:start position:0% worth it again this is all pro time the 00:16:31.360 --> 00:16:33.629 align:start position:0% worth it again this is all pro time the one<00:16:31.560> thing<00:16:31.759> that<00:16:32.000> this<00:16:32.720> slide<00:16:33.079> doesn't<00:16:33.399> talk 00:16:33.629 --> 00:16:33.639 align:start position:0% one thing that this slide doesn't talk 00:16:33.639 --> 00:16:37.389 align:start position:0% one thing that this slide doesn't talk about<00:16:34.160> is<00:16:35.160> uh<00:16:35.600> proof<00:16:36.000> size<00:16:36.240> and<00:16:36.399> verifier 00:16:37.389 --> 00:16:37.399 align:start position:0% about is uh proof size and verifier 00:16:37.399 --> 00:16:42.389 align:start position:0% about is uh proof size and verifier time<00:16:38.399> maybe<00:16:39.160> the<00:16:40.160> the<00:16:40.319> big<00:16:40.839> the<00:16:40.959> big<00:16:41.399> drawback 00:16:42.389 --> 00:16:42.399 align:start position:0% time maybe the the big the big drawback 00:16:42.399 --> 00:16:45.870 align:start position:0% time maybe the the big the big drawback uh<00:16:42.600> for<00:16:43.040> log<00:16:43.279> up<00:16:43.399> gkr<00:16:43.920> I'd<00:16:44.079> say<00:16:44.360> is<00:16:44.519> proof<00:16:44.880> size 00:16:45.870 --> 00:16:45.880 align:start position:0% uh for log up gkr I'd say is proof size 00:16:45.880 --> 00:16:48.430 align:start position:0% uh for log up gkr I'd say is proof size so<00:16:46.639> these<00:16:46.839> extra<00:16:47.160> columns<00:16:47.519> that<00:16:47.600> we're<00:16:47.800> adding 00:16:48.430 --> 00:16:48.440 align:start position:0% so these extra columns that we're adding 00:16:48.440 --> 00:16:51.189 align:start position:0% so these extra columns that we're adding and<00:16:48.920> all<00:16:49.160> the<00:16:49.399> nodes<00:16:50.199> that<00:16:50.560> the<00:16:50.839> the<00:16:51.120> the 00:16:51.189 --> 00:16:51.199 align:start position:0% and all the nodes that the the the 00:16:51.199 --> 00:16:55.389 align:start position:0% and all the nodes that the the the circuit<00:16:51.519> nodes<00:16:51.839> for<00:16:52.279> log<00:16:52.519> up<00:16:52.800> gkr<00:16:53.800> are<00:16:54.480> um<00:16:55.240> are 00:16:55.389 --> 00:16:55.399 align:start position:0% circuit nodes for log up gkr are um are 00:16:55.399 --> 00:16:58.470 align:start position:0% circuit nodes for log up gkr are um are all<00:16:55.560> in<00:16:55.680> the<00:16:55.920> in<00:16:56.040> the<00:16:56.199> extension<00:16:56.600> field<00:16:57.480> so 00:16:58.470 --> 00:16:58.480 align:start position:0% all in the in the extension field so 00:16:58.480 --> 00:17:02.230 align:start position:0% all in the in the extension field so example<00:16:58.959> but<00:16:59.440> uh<00:16:59.639> in<00:16:59.759> the<00:16:59.839> M<00:17:00.079> VM<00:17:00.440> we<00:17:00.959> use<00:17:01.959> uh 00:17:02.230 --> 00:17:02.240 align:start position:0% example but uh in the M VM we use uh 00:17:02.240 --> 00:17:05.390 align:start position:0% example but uh in the M VM we use uh goldilock<00:17:02.880> so<00:17:03.319> roughly<00:17:03.639> 64-bit<00:17:04.400> field 00:17:05.390 --> 00:17:05.400 align:start position:0% goldilock so roughly 64-bit field 00:17:05.400 --> 00:17:07.990 align:start position:0% goldilock so roughly 64-bit field extension<00:17:05.799> field<00:17:06.199> is<00:17:06.679> uh<00:17:07.199> where<00:17:07.319> we<00:17:07.439> do<00:17:07.679> any 00:17:07.990 --> 00:17:08.000 align:start position:0% extension field is uh where we do any 00:17:08.000 --> 00:17:09.270 align:start position:0% extension field is uh where we do any any<00:17:08.160> kind<00:17:08.240> of<00:17:08.360> Randomness<00:17:08.799> is<00:17:08.919> done<00:17:09.079> in<00:17:09.160> the 00:17:09.270 --> 00:17:09.280 align:start position:0% any kind of Randomness is done in the 00:17:09.280 --> 00:17:11.710 align:start position:0% any kind of Randomness is done in the extension<00:17:09.679> field<00:17:10.160> and<00:17:10.319> that's<00:17:10.679> roughly<00:17:11.000> 128 00:17:11.710 --> 00:17:11.720 align:start position:0% extension field and that's roughly 128 00:17:11.720 --> 00:17:14.990 align:start position:0% extension field and that's roughly 128 bits<00:17:12.400> so<00:17:12.640> every<00:17:12.919> number<00:17:13.280> is<00:17:13.520> is<00:17:13.679> larger<00:17:14.679> and 00:17:14.990 --> 00:17:15.000 align:start position:0% bits so every number is is larger and 00:17:15.000 --> 00:17:19.829 align:start position:0% bits so every number is is larger and proofs<00:17:15.720> kind<00:17:15.839> of<00:17:16.600> at<00:17:16.880> very<00:17:17.120> large<00:17:18.079> uh<00:17:18.839> for 00:17:19.829 --> 00:17:19.839 align:start position:0% proofs kind of at very large uh for 00:17:19.839 --> 00:17:23.110 align:start position:0% proofs kind of at very large uh for traces<00:17:20.600> that<00:17:20.760> are<00:17:21.720> that<00:17:21.839> we<00:17:21.959> can<00:17:22.199> expect<00:17:22.559> in<00:17:22.959> in 00:17:23.110 --> 00:17:23.120 align:start position:0% traces that are that we can expect in in 00:17:23.120 --> 00:17:25.350 align:start position:0% traces that are that we can expect in in the<00:17:23.240> rollup<00:17:23.919> we<00:17:24.039> were<00:17:24.240> getting<00:17:24.559> maybe<00:17:25.039> 200 00:17:25.350 --> 00:17:25.360 align:start position:0% the rollup we were getting maybe 200 00:17:25.360 --> 00:17:29.150 align:start position:0% the rollup we were getting maybe 200 kilobyte<00:17:26.039> size<00:17:26.640> uh<00:17:26.760> proofs<00:17:27.160> just<00:17:27.319> for<00:17:27.640> gkr<00:17:28.640> and 00:17:29.150 --> 00:17:29.160 align:start position:0% kilobyte size uh proofs just for gkr and 00:17:29.160 --> 00:17:31.230 align:start position:0% kilobyte size uh proofs just for gkr and typically<00:17:29.880> we're<00:17:30.120> able<00:17:30.320> to<00:17:30.480> get<00:17:30.600> it<00:17:30.840> maybe<00:17:31.080> at 00:17:31.230 --> 00:17:31.240 align:start position:0% typically we're able to get it maybe at 00:17:31.240 --> 00:17:32.669 align:start position:0% typically we're able to get it maybe at 80<00:17:31.559> or<00:17:31.880> 100 00:17:32.669 --> 00:17:32.679 align:start position:0% 80 or 100 00:17:32.679 --> 00:17:35.150 align:start position:0% 80 or 100 kilobytes<00:17:33.679> uh<00:17:33.840> so<00:17:34.080> it<00:17:34.280> really<00:17:34.480> blew<00:17:34.799> up<00:17:35.000> our 00:17:35.150 --> 00:17:35.160 align:start position:0% kilobytes uh so it really blew up our 00:17:35.160 --> 00:17:37.549 align:start position:0% kilobytes uh so it really blew up our our<00:17:35.320> proof<00:17:35.600> sizes<00:17:36.039> up<00:17:37.039> and<00:17:37.160> we're<00:17:37.360> actually 00:17:37.549 --> 00:17:37.559 align:start position:0% our proof sizes up and we're actually 00:17:37.559 --> 00:17:41.430 align:start position:0% our proof sizes up and we're actually trying<00:17:37.760> to<00:17:38.280> to<00:17:38.480> decrease<00:17:38.880> them<00:17:39.080> so<00:17:39.400> that<00:17:40.400> uh<00:17:41.120> so 00:17:41.430 --> 00:17:41.440 align:start position:0% trying to to decrease them so that uh so 00:17:41.440 --> 00:17:42.990 align:start position:0% trying to to decrease them so that uh so ultimately 00:17:42.990 --> 00:17:43.000 align:start position:0% ultimately 00:17:43.000 --> 00:17:47.070 align:start position:0% ultimately our<00:17:44.000> our<00:17:44.280> current<00:17:45.360> um<00:17:46.360> takeaway<00:17:46.840> is<00:17:46.960> that 00:17:47.070 --> 00:17:47.080 align:start position:0% our our current um takeaway is that 00:17:47.080 --> 00:17:48.510 align:start position:0% our our current um takeaway is that we're<00:17:47.240> not<00:17:47.360> going<00:17:47.440> to<00:17:47.559> use<00:17:47.720> it<00:17:47.880> for<00:17:48.080> the<00:17:48.240> first 00:17:48.510 --> 00:17:48.520 align:start position:0% we're not going to use it for the first 00:17:48.520 --> 00:17:51.150 align:start position:0% we're not going to use it for the first version<00:17:48.799> of<00:17:49.160> of<00:17:49.280> the<00:17:49.360> M<00:17:49.640> VM<00:17:50.480> but<00:17:50.600> we're<00:17:50.880> also 00:17:51.150 --> 00:17:51.160 align:start position:0% version of of the M VM but we're also 00:17:51.160 --> 00:17:53.470 align:start position:0% version of of the M VM but we're also looking<00:17:51.480> into<00:17:52.200> uh<00:17:52.320> making<00:17:52.559> some<00:17:53.200> pretty 00:17:53.470 --> 00:17:53.480 align:start position:0% looking into uh making some pretty 00:17:53.480 --> 00:17:56.430 align:start position:0% looking into uh making some pretty drastic<00:17:53.880> changes<00:17:54.400> in<00:17:54.520> the<00:17:54.799> future<00:17:55.799> and<00:17:56.240> uh 00:17:56.430 --> 00:17:56.440 align:start position:0% drastic changes in the future and uh 00:17:56.440 --> 00:17:57.750 align:start position:0% drastic changes in the future and uh yeah<00:17:56.600> this<00:17:56.720> is<00:17:56.840> the<00:17:56.960> m<00:17:57.200> map<00:17:57.400> we're<00:17:57.520> going<00:17:57.600> to 00:17:57.750 --> 00:17:57.760 align:start position:0% yeah this is the m map we're going to 00:17:57.760 --> 00:18:01.750 align:start position:0% yeah this is the m map we're going to use<00:17:58.559> uh<00:17:59.039> when<00:17:59.320> we<00:18:00.320> when<00:18:00.440> we<00:18:00.600> do<00:18:00.840> that<00:18:01.240> and 00:18:01.750 --> 00:18:01.760 align:start position:0% use uh when we when we do that and 00:18:01.760 --> 00:18:04.789 align:start position:0% use uh when we when we do that and whether<00:18:01.960> or<00:18:02.120> not<00:18:02.280> we<00:18:02.440> want<00:18:02.640> to<00:18:03.360> use<00:18:03.559> log<00:18:03.760> of<00:18:03.880> gkr 00:18:04.789 --> 00:18:04.799 align:start position:0% whether or not we want to use log of gkr 00:18:04.799 --> 00:18:06.950 align:start position:0% whether or not we want to use log of gkr But<00:18:05.280> ultimately<00:18:05.679> it<00:18:05.760> was<00:18:06.080> a<00:18:06.400> a<00:18:06.600> great 00:18:06.950 --> 00:18:06.960 align:start position:0% But ultimately it was a a great 00:18:06.960 --> 00:18:09.470 align:start position:0% But ultimately it was a a great experience<00:18:07.480> and<00:18:07.880> uh<00:18:08.559> like<00:18:08.720> a<00:18:08.880> very<00:18:09.120> clever 00:18:09.470 --> 00:18:09.480 align:start position:0% experience and uh like a very clever 00:18:09.480 --> 00:18:12.190 align:start position:0% experience and uh like a very clever idea<00:18:10.000> by<00:18:10.320> Shahar<00:18:10.880> and 00:18:12.190 --> 00:18:12.200 align:start position:0% idea by Shahar and 00:18:12.200 --> 00:18:15.909 align:start position:0% idea by Shahar and O<00:18:13.200> so<00:18:13.480> again<00:18:13.760> like<00:18:14.159> most<00:18:14.320> of<00:18:14.440> the<00:18:14.600> de<00:18:14.919> details 00:18:15.909 --> 00:18:15.919 align:start position:0% O so again like most of the de details 00:18:15.919 --> 00:18:18.549 align:start position:0% O so again like most of the de details uh<00:18:16.440> we<00:18:16.559> didn't<00:18:16.799> talk<00:18:17.039> about<00:18:17.559> if<00:18:17.720> you<00:18:18.200> go<00:18:18.360> to 00:18:18.549 --> 00:18:18.559 align:start position:0% uh we didn't talk about if you go to 00:18:18.559 --> 00:18:21.470 align:start position:0% uh we didn't talk about if you go to that<00:18:18.760> link<00:18:19.320> uh<00:18:19.520> most<00:18:19.720> of<00:18:19.880> them<00:18:20.000> are<00:18:20.240> there<00:18:21.120> I<00:18:21.360> I 00:18:21.470 --> 00:18:21.480 align:start position:0% that link uh most of them are there I I 00:18:21.480 --> 00:18:24.029 align:start position:0% that link uh most of them are there I I try<00:18:21.760> and<00:18:22.000> make<00:18:22.159> it<00:18:22.720> approachable<00:18:23.720> uh<00:18:23.840> feel 00:18:24.029 --> 00:18:24.039 align:start position:0% try and make it approachable uh feel 00:18:24.039 --> 00:18:26.230 align:start position:0% try and make it approachable uh feel free<00:18:24.240> to<00:18:24.440> ask<00:18:24.679> questions<00:18:25.240> on<00:18:25.400> the<00:18:25.559> hackmd 00:18:26.230 --> 00:18:26.240 align:start position:0% free to ask questions on the hackmd 00:18:26.240 --> 00:18:29.230 align:start position:0% free to ask questions on the hackmd directly<00:18:27.240> uh<00:18:27.440> send<00:18:27.640> me<00:18:27.799> a<00:18:27.880> DM<00:18:28.159> on<00:18:28.280> Twitter<00:18:28.799> or 00:18:29.230 --> 00:18:29.240 align:start position:0% directly uh send me a DM on Twitter or 00:18:29.240 --> 00:18:32.230 align:start position:0% directly uh send me a DM on Twitter or just<00:18:29.440> message<00:18:29.760> me<00:18:30.480> uh<00:18:31.400> whichever<00:18:31.799> I'm<00:18:31.960> happy 00:18:32.230 --> 00:18:32.240 align:start position:0% just message me uh whichever I'm happy 00:18:32.240 --> 00:18:34.830 align:start position:0% just message me uh whichever I'm happy to<00:18:32.720> to<00:18:33.400> have<00:18:33.559> more<00:18:33.760> conversations<00:18:34.320> about<00:18:34.600> this 00:18:34.830 --> 00:18:34.840 align:start position:0% to to have more conversations about this 00:18:34.840 --> 00:18:37.590 align:start position:0% to to have more conversations about this and<00:18:35.760> yeah<00:18:36.280> uh<00:18:36.440> hopefully<00:18:36.880> youn<00:18:37.240> you<00:18:37.360> learned 00:18:37.590 --> 00:18:37.600 align:start position:0% and yeah uh hopefully youn you learned 00:18:37.600 --> 00:18:42.350 align:start position:0% and yeah uh hopefully youn you learned something<00:18:38.159> thank 00:18:42.350 --> 00:18:42.360 align:start position:0% 00:18:42.360 --> 00:18:44.909 align:start position:0% you<00:18:43.360> thank<00:18:43.480> you<00:18:43.640> phip<00:18:44.000> it<00:18:44.120> was<00:18:44.240> an<00:18:44.360> insightful 00:18:44.909 --> 00:18:44.919 align:start position:0% you thank you phip it was an insightful 00:18:44.919 --> 00:18:47.350 align:start position:0% you thank you phip it was an insightful talk<00:18:45.919> so<00:18:46.200> again<00:18:46.400> I<00:18:46.559> just<00:18:46.760> remind<00:18:47.080> you<00:18:47.200> that 00:18:47.350 --> 00:18:47.360 align:start position:0% talk so again I just remind you that 00:18:47.360 --> 00:18:49.510 align:start position:0% talk so again I just remind you that there<00:18:47.440> is<00:18:47.559> a<00:18:47.679> q<00:18:48.000> code<00:18:48.159> if<00:18:48.240> you<00:18:48.320> want<00:18:48.440> to<00:18:48.600> ask 00:18:49.510 --> 00:18:49.520 align:start position:0% there is a q code if you want to ask 00:18:49.520 --> 00:18:53.990 align:start position:0% there is a q code if you want to ask questions<00:18:50.520> um<00:18:51.200> but<00:18:51.400> we<00:18:51.600> have<00:18:51.960> actually<00:18:52.600> a<00:18:53.000> few 00:18:53.990 --> 00:18:54.000 align:start position:0% questions um but we have actually a few 00:18:54.000 --> 00:18:56.830 align:start position:0% questions um but we have actually a few so<00:18:54.840> maybe<00:18:55.320> why<00:18:55.440> if<00:18:55.559> we<00:18:55.679> start<00:18:55.960> with<00:18:56.120> the<00:18:56.240> first 00:18:56.830 --> 00:18:56.840 align:start position:0% so maybe why if we start with the first 00:18:56.840 --> 00:18:59.270 align:start position:0% so maybe why if we start with the first one<00:18:57.840> do<00:18:58.200> we 00:18:59.270 --> 00:18:59.280 align:start position:0% one do we 00:18:59.280 --> 00:19:01.630 align:start position:0% one do we uh<00:18:59.400> in<00:18:59.640> practice<00:19:00.039> mix<00:19:00.400> JK<00:19:00.880> and<00:19:01.039> stock<00:19:01.400> for 00:19:01.630 --> 00:19:01.640 align:start position:0% uh in practice mix JK and stock for 00:19:01.640 --> 00:19:03.470 align:start position:0% uh in practice mix JK and stock for different<00:19:01.960> use<00:19:02.240> cases<00:19:02.600> other<00:19:02.840> than<00:19:03.000> look<00:19:03.200> up<00:19:03.320> I 00:19:03.470 --> 00:19:03.480 align:start position:0% different use cases other than look up I 00:19:03.480 --> 00:19:05.870 align:start position:0% different use cases other than look up I guess<00:19:04.320> yeah<00:19:04.520> that's<00:19:04.640> a<00:19:04.799> really<00:19:05.000> good<00:19:05.200> question 00:19:05.870 --> 00:19:05.880 align:start position:0% guess yeah that's a really good question 00:19:05.880 --> 00:19:07.510 align:start position:0% guess yeah that's a really good question uh<00:19:06.000> and<00:19:06.159> something<00:19:06.440> that<00:19:06.559> we<00:19:06.760> start<00:19:07.000> to<00:19:07.280> think 00:19:07.510 --> 00:19:07.520 align:start position:0% uh and something that we start to think 00:19:07.520 --> 00:19:10.590 align:start position:0% uh and something that we start to think about<00:19:07.919> more<00:19:08.919> uh<00:19:09.480> currently<00:19:09.840> log<00:19:10.080> up<00:19:10.280> is<00:19:10.440> the 00:19:10.590 --> 00:19:10.600 align:start position:0% about more uh currently log up is the 00:19:10.600 --> 00:19:12.789 align:start position:0% about more uh currently log up is the only<00:19:10.840> one<00:19:11.039> that<00:19:11.200> I'm<00:19:11.440> aware<00:19:11.720> of<00:19:11.960> that<00:19:12.120> is<00:19:12.559> being 00:19:12.789 --> 00:19:12.799 align:start position:0% only one that I'm aware of that is being 00:19:12.799 --> 00:19:17.190 align:start position:0% only one that I'm aware of that is being used<00:19:13.120> or<00:19:13.280> even<00:19:13.679> tried<00:19:14.760> um<00:19:15.760> but<00:19:16.360> uh<00:19:16.679> it<00:19:16.799> is<00:19:16.960> on 00:19:17.190 --> 00:19:17.200 align:start position:0% used or even tried um but uh it is on 00:19:17.200 --> 00:19:19.950 align:start position:0% used or even tried um but uh it is on our<00:19:18.159> we<00:19:18.280> are<00:19:18.480> going<00:19:18.600> to<00:19:18.760> think<00:19:19.000> about<00:19:19.640> more 00:19:19.950 --> 00:19:19.960 align:start position:0% our we are going to think about more 00:19:19.960 --> 00:19:22.110 align:start position:0% our we are going to think about more ways<00:19:20.679> uh<00:19:20.840> that<00:19:20.919> we<00:19:21.039> can<00:19:21.159> do<00:19:21.360> this<00:19:21.520> because<00:19:21.840> now 00:19:22.110 --> 00:19:22.120 align:start position:0% ways uh that we can do this because now 00:19:22.120 --> 00:19:24.510 align:start position:0% ways uh that we can do this because now we<00:19:22.480> we<00:19:22.640> have<00:19:22.840> a<00:19:23.000> clear<00:19:23.360> use<00:19:23.720> case<00:19:24.000> and<00:19:24.200> or 00:19:24.510 --> 00:19:24.520 align:start position:0% we we have a clear use case and or 00:19:24.520 --> 00:19:26.430 align:start position:0% we we have a clear use case and or rather<00:19:24.720> a<00:19:24.840> clear<00:19:25.159> example<00:19:25.640> of<00:19:25.840> how<00:19:26.000> it<00:19:26.159> can 00:19:26.430 --> 00:19:26.440 align:start position:0% rather a clear example of how it can 00:19:26.440 --> 00:19:29.830 align:start position:0% rather a clear example of how it can work<00:19:27.320> and<00:19:27.760> uh<00:19:28.000> yeah<00:19:28.400> but<00:19:28.720> so<00:19:28.880> far<00:19:29.120> I<00:19:29.320> I<00:19:29.559> don't 00:19:29.830 --> 00:19:29.840 align:start position:0% work and uh yeah but so far I I don't 00:19:29.840 --> 00:19:31.470 align:start position:0% work and uh yeah but so far I I don't think<00:19:30.000> so<00:19:30.360> so<00:19:30.520> there<00:19:30.679> Reas<00:19:30.880> is<00:19:31.039> like<00:19:31.320> just 00:19:31.470 --> 00:19:31.480 align:start position:0% think so so there Reas is like just 00:19:31.480 --> 00:19:33.190 align:start position:0% think so so there Reas is like just implementation<00:19:32.159> wise<00:19:32.360> you<00:19:32.440> need<00:19:32.559> to<00:19:32.760> try<00:19:33.039> it 00:19:33.190 --> 00:19:33.200 align:start position:0% implementation wise you need to try it 00:19:33.200 --> 00:19:35.830 align:start position:0% implementation wise you need to try it to<00:19:33.360> see<00:19:34.120> how<00:19:34.280> it<00:19:34.440> works<00:19:35.039> so<00:19:35.240> implementation 00:19:35.830 --> 00:19:35.840 align:start position:0% to see how it works so implementation 00:19:35.840 --> 00:19:38.190 align:start position:0% to see how it works so implementation wise<00:19:36.120> absolutely<00:19:36.880> even<00:19:37.280> just<00:19:37.799> understanding 00:19:38.190 --> 00:19:38.200 align:start position:0% wise absolutely even just understanding 00:19:38.200 --> 00:19:40.510 align:start position:0% wise absolutely even just understanding like<00:19:38.320> the<00:19:38.480> glue<00:19:38.840> from<00:19:39.240> from<00:19:39.400> gkr<00:19:39.840> to<00:19:40.000> Starks 00:19:40.510 --> 00:19:40.520 align:start position:0% like the glue from from gkr to Starks 00:19:40.520 --> 00:19:42.310 align:start position:0% like the glue from from gkr to Starks was<00:19:41.000> non-trivial<00:19:41.600> to<00:19:41.919> understand<00:19:42.080> at<00:19:42.159> least 00:19:42.310 --> 00:19:42.320 align:start position:0% was non-trivial to understand at least 00:19:42.320 --> 00:19:43.510 align:start position:0% was non-trivial to understand at least for<00:19:42.480> me<00:19:42.760> because<00:19:42.880> you're<00:19:43.000> going<00:19:43.240> from<00:19:43.360> a 00:19:43.510 --> 00:19:43.520 align:start position:0% for me because you're going from a 00:19:43.520 --> 00:19:46.230 align:start position:0% for me because you're going from a multi-linear<00:19:44.200> world<00:19:44.440> to<00:19:44.559> a<00:19:45.120> uh<00:19:45.720> univariant 00:19:46.230 --> 00:19:46.240 align:start position:0% multi-linear world to a uh univariant 00:19:46.240 --> 00:19:49.070 align:start position:0% multi-linear world to a uh univariant world<00:19:47.080> uh<00:19:47.240> and<00:19:47.360> there's<00:19:47.600> some<00:19:48.520> but<00:19:48.720> I'm<00:19:48.880> pretty 00:19:49.070 --> 00:19:49.080 align:start position:0% world uh and there's some but I'm pretty 00:19:49.080 --> 00:19:50.590 align:start position:0% world uh and there's some but I'm pretty sure<00:19:49.320> that<00:19:49.440> some<00:19:49.559> of<00:19:49.720> the<00:19:49.880> insights<00:19:50.360> there<00:19:50.480> can 00:19:50.590 --> 00:19:50.600 align:start position:0% sure that some of the insights there can 00:19:50.600 --> 00:19:53.909 align:start position:0% sure that some of the insights there can be<00:19:50.720> reused<00:19:51.120> for<00:19:51.280> other<00:19:51.559> purposes<00:19:52.559> that<00:19:52.760> makes 00:19:53.909 --> 00:19:53.919 align:start position:0% be reused for other purposes that makes 00:19:53.919 --> 00:19:57.390 align:start position:0% be reused for other purposes that makes sense<00:19:54.919> um<00:19:55.919> yeah<00:19:56.240> why<00:19:56.400> if<00:19:56.520> we<00:19:56.679> take<00:19:56.880> the<00:19:57.039> second 00:19:57.390 --> 00:19:57.400 align:start position:0% sense um yeah why if we take the second 00:19:57.400 --> 00:20:00.470 align:start position:0% sense um yeah why if we take the second question<00:19:57.880> is<00:19:58.159> log<00:19:58.440> up<00:19:59.120> more<00:19:59.440> gkr<00:20:00.039> friendly 00:20:00.470 --> 00:20:00.480 align:start position:0% question is log up more gkr friendly 00:20:00.480 --> 00:20:02.070 align:start position:0% question is log up more gkr friendly than<00:20:00.720> other<00:20:00.960> lookup 00:20:02.070 --> 00:20:02.080 align:start position:0% than other lookup 00:20:02.080 --> 00:20:04.830 align:start position:0% than other lookup arguments<00:20:03.080> yeah<00:20:03.280> another<00:20:03.600> good<00:20:03.840> question<00:20:04.559> uh 00:20:04.830 --> 00:20:04.840 align:start position:0% arguments yeah another good question uh 00:20:04.840 --> 00:20:08.590 align:start position:0% arguments yeah another good question uh I<00:20:04.960> do<00:20:05.120> not<00:20:05.360> know<00:20:05.720> any<00:20:06.240> other<00:20:07.240> uses<00:20:07.679> of<00:20:08.400> there 00:20:08.590 --> 00:20:08.600 align:start position:0% I do not know any other uses of there 00:20:08.600 --> 00:20:11.390 align:start position:0% I do not know any other uses of there might<00:20:08.799> be<00:20:09.000> but<00:20:09.159> I<00:20:09.240> do<00:20:09.400> not<00:20:09.559> know<00:20:10.240> uh<00:20:10.679> other 00:20:11.390 --> 00:20:11.400 align:start position:0% might be but I do not know uh other 00:20:11.400 --> 00:20:15.750 align:start position:0% might be but I do not know uh other lookup<00:20:11.760> Arguments<00:20:12.559> for<00:20:13.320> uh<00:20:13.480> that<00:20:13.640> use<00:20:13.840> gkr<00:20:14.760> so 00:20:15.750 --> 00:20:15.760 align:start position:0% lookup Arguments for uh that use gkr so 00:20:15.760 --> 00:20:16.990 align:start position:0% lookup Arguments for uh that use gkr so uh<00:20:15.919> I<00:20:16.080> I<00:20:16.159> don't 00:20:16.990 --> 00:20:17.000 align:start position:0% uh I I don't 00:20:17.000 --> 00:20:19.310 align:start position:0% uh I I don't know<00:20:18.000> and<00:20:18.120> you<00:20:18.240> were<00:20:18.400> using<00:20:18.679> like<00:20:18.840> log<00:20:19.080> UPS 00:20:19.310 --> 00:20:19.320 align:start position:0% know and you were using like log UPS 00:20:19.320 --> 00:20:21.950 align:start position:0% know and you were using like log UPS before<00:20:19.600> introducing<00:20:20.280> JKR<00:20:20.840> so<00:20:21.120> we<00:20:21.480> it<00:20:21.640> was 00:20:21.950 --> 00:20:21.960 align:start position:0% before introducing JKR so we it was 00:20:21.960 --> 00:20:25.310 align:start position:0% before introducing JKR so we it was stock<00:20:22.320> friendly<00:20:22.760> I<00:20:22.960> guess<00:20:23.600> right<00:20:23.960> so<00:20:24.640> we<00:20:24.840> use 00:20:25.310 --> 00:20:25.320 align:start position:0% stock friendly I guess right so we use 00:20:25.320 --> 00:20:27.630 align:start position:0% stock friendly I guess right so we use actually<00:20:25.559> we<00:20:25.679> use<00:20:25.919> log<00:20:26.200> up<00:20:26.480> only<00:20:26.720> when<00:20:26.880> we<00:20:27.000> need 00:20:27.630 --> 00:20:27.640 align:start position:0% actually we use log up only when we need 00:20:27.640 --> 00:20:29.950 align:start position:0% actually we use log up only when we need multiplicities<00:20:28.720> because<00:20:29.159> uh<00:20:29.320> as<00:20:29.440> we<00:20:29.600> saw<00:20:29.840> it 00:20:29.950 --> 00:20:29.960 align:start position:0% multiplicities because uh as we saw it 00:20:29.960 --> 00:20:32.590 align:start position:0% multiplicities because uh as we saw it makes<00:20:30.120> the<00:20:30.280> degree<00:20:31.039> log<00:20:31.280> up<00:20:31.480> makes<00:20:31.679> the<00:20:31.799> degree 00:20:32.590 --> 00:20:32.600 align:start position:0% makes the degree log up makes the degree 00:20:32.600 --> 00:20:35.110 align:start position:0% makes the degree log up makes the degree go<00:20:33.240> uh<00:20:33.760> go<00:20:33.960> up<00:20:34.159> pretty<00:20:34.400> quickly<00:20:34.760> because<00:20:34.960> of 00:20:35.110 --> 00:20:35.120 align:start position:0% go uh go up pretty quickly because of 00:20:35.120 --> 00:20:36.510 align:start position:0% go uh go up pretty quickly because of these<00:20:35.280> denominators<00:20:35.799> you<00:20:35.919> have<00:20:36.000> to<00:20:36.120> multiply 00:20:36.510 --> 00:20:36.520 align:start position:0% these denominators you have to multiply 00:20:36.520 --> 00:20:39.549 align:start position:0% these denominators you have to multiply out<00:20:37.320> I<00:20:37.400> mean<00:20:37.559> a<00:20:37.679> constraint<00:20:38.039> degree<00:20:38.400> sorry<00:20:39.320> uh 00:20:39.549 --> 00:20:39.559 align:start position:0% out I mean a constraint degree sorry uh 00:20:39.559 --> 00:20:41.310 align:start position:0% out I mean a constraint degree sorry uh for<00:20:39.799> most<00:20:39.960> of<00:20:40.080> our<00:20:40.240> lookup<00:20:40.600> arguments<00:20:41.159> in 00:20:41.310 --> 00:20:41.320 align:start position:0% for most of our lookup arguments in 00:20:41.320 --> 00:20:43.870 align:start position:0% for most of our lookup arguments in Starks<00:20:41.799> we<00:20:41.919> use<00:20:42.600> good<00:20:42.760> old<00:20:43.000> multi<00:20:43.400> set<00:20:43.520> checks 00:20:43.870 --> 00:20:43.880 align:start position:0% Starks we use good old multi set checks 00:20:43.880 --> 00:20:45.590 align:start position:0% Starks we use good old multi set checks we<00:20:44.120> I<00:20:44.200> call<00:20:44.360> them<00:20:44.520> like<00:20:44.720> vanilla<00:20:45.120> multi<00:20:45.480> set 00:20:45.590 --> 00:20:45.600 align:start position:0% we I call them like vanilla multi set 00:20:45.600 --> 00:20:48.310 align:start position:0% we I call them like vanilla multi set checks<00:20:46.000> they're<00:20:46.679> based<00:20:47.000> on<00:20:47.679> they're<00:20:47.840> not<00:20:48.039> sums 00:20:48.310 --> 00:20:48.320 align:start position:0% checks they're based on they're not sums 00:20:48.320 --> 00:20:51.270 align:start position:0% checks they're based on they're not sums of<00:20:48.520> fractions<00:20:48.960> they're<00:20:49.159> really<00:20:50.080> uh<00:20:51.080> it's<00:20:51.159> not 00:20:51.270 --> 00:20:51.280 align:start position:0% of fractions they're really uh it's not 00:20:51.280 --> 00:20:52.830 align:start position:0% of fractions they're really uh it's not a<00:20:51.360> running<00:20:51.640> sum<00:20:51.960> column<00:20:52.200> it's<00:20:52.280> a<00:20:52.400> running 00:20:52.830 --> 00:20:52.840 align:start position:0% a running sum column it's a running 00:20:52.840 --> 00:20:54.430 align:start position:0% a running sum column it's a running product<00:20:53.159> and<00:20:53.280> it's<00:20:53.440> slightly<00:20:53.840> different<00:20:54.240> your 00:20:54.430 --> 00:20:54.440 align:start position:0% product and it's slightly different your 00:20:54.440 --> 00:20:56.190 align:start position:0% product and it's slightly different your degree<00:20:54.799> is<00:20:54.840> slightly<00:20:55.200> better<00:20:55.679> so<00:20:55.840> unless<00:20:56.120> you 00:20:56.190 --> 00:20:56.200 align:start position:0% degree is slightly better so unless you 00:20:56.200 --> 00:20:58.750 align:start position:0% degree is slightly better so unless you need<00:20:56.400> to<00:20:56.559> uncode<00:20:57.000> multiplicities<00:20:57.760> explicitly 00:20:58.750 --> 00:20:58.760 align:start position:0% need to uncode multiplicities explicitly 00:20:58.760 --> 00:21:00.430 align:start position:0% need to uncode multiplicities explicitly uh<00:20:58.880> we<00:20:59.000> don't<00:20:59.159> use<00:20:59.360> log<00:20:59.600> up<00:20:59.960> but<00:21:00.120> I<00:21:00.159> don't<00:21:00.320> know 00:21:00.430 --> 00:21:00.440 align:start position:0% uh we don't use log up but I don't know 00:21:00.440 --> 00:21:02.510 align:start position:0% uh we don't use log up but I don't know how<00:21:00.520> to<00:21:00.640> do<00:21:00.799> that<00:21:00.960> other<00:21:01.159> one<00:21:01.280> with<00:21:01.400> JR<00:21:02.320> okay 00:21:02.510 --> 00:21:02.520 align:start position:0% how to do that other one with JR okay 00:21:02.520 --> 00:21:03.350 align:start position:0% how to do that other one with JR okay cool 00:21:03.350 --> 00:21:03.360 align:start position:0% cool 00:21:03.360 --> 00:21:07.029 align:start position:0% cool cool<00:21:04.360> we<00:21:04.480> have<00:21:04.679> more<00:21:05.039> questions<00:21:06.039> so<00:21:06.760> what's 00:21:07.029 --> 00:21:07.039 align:start position:0% cool we have more questions so what's 00:21:07.039 --> 00:21:09.270 align:start position:0% cool we have more questions so what's hash<00:21:07.360> function<00:21:07.799> function<00:21:08.159> you<00:21:08.240> are<00:21:08.440> using 00:21:09.270 --> 00:21:09.280 align:start position:0% hash function function you are using 00:21:09.280 --> 00:21:13.350 align:start position:0% hash function function you are using right<00:21:09.480> so<00:21:09.840> the<00:21:10.640> M<00:21:10.880> VM<00:21:11.360> does<00:21:12.360> does<00:21:12.600> not<00:21:13.120> you<00:21:13.200> can 00:21:13.350 --> 00:21:13.360 align:start position:0% right so the M VM does does not you can 00:21:13.360 --> 00:21:15.950 align:start position:0% right so the M VM does does not you can use<00:21:13.600> whichever<00:21:13.919> one<00:21:14.039> you<00:21:14.159> want<00:21:14.320> with<00:21:14.440> the<00:21:14.600> m<00:21:14.960> VM 00:21:15.950 --> 00:21:15.960 align:start position:0% use whichever one you want with the m VM 00:21:15.960 --> 00:21:18.750 align:start position:0% use whichever one you want with the m VM uh<00:21:16.640> we<00:21:16.799> use<00:21:17.000> the<00:21:17.080> M<00:21:17.320> VM<00:21:17.559> in<00:21:17.679> the<00:21:17.760> M<00:21:18.039> rollup<00:21:18.600> and 00:21:18.750 --> 00:21:18.760 align:start position:0% uh we use the M VM in the M rollup and 00:21:18.760 --> 00:21:21.669 align:start position:0% uh we use the M VM in the M rollup and for<00:21:18.960> that<00:21:19.159> we<00:21:19.320> use<00:21:19.880> uh<00:21:20.039> hash<00:21:20.279> function<00:21:20.679> RPO 00:21:21.669 --> 00:21:21.679 align:start position:0% for that we use uh hash function RPO 00:21:21.679 --> 00:21:25.510 align:start position:0% for that we use uh hash function RPO which<00:21:21.840> is<00:21:22.200> uh<00:21:23.039> ZK<00:21:23.440> friendly<00:21:23.960> hash<00:21:24.520> function 00:21:25.510 --> 00:21:25.520 align:start position:0% which is uh ZK friendly hash function 00:21:25.520 --> 00:21:26.590 align:start position:0% which is uh ZK friendly hash function and<00:21:25.679> we<00:21:25.840> actually<00:21:26.000> I<00:21:26.080> think<00:21:26.200> we're<00:21:26.320> going<00:21:26.440> to 00:21:26.590 --> 00:21:26.600 align:start position:0% and we actually I think we're going to 00:21:26.600 --> 00:21:30.070 align:start position:0% and we actually I think we're going to switch<00:21:26.880> soon<00:21:27.120> to<00:21:27.279> RPX<00:21:28.120> so<00:21:28.720> is<00:21:28.840> it<00:21:29.000> also<00:21:29.919> it's 00:21:30.070 --> 00:21:30.080 align:start position:0% switch soon to RPX so is it also it's 00:21:30.080 --> 00:21:31.750 align:start position:0% switch soon to RPX so is it also it's like<00:21:30.279> an<00:21:30.480> improvement<00:21:31.240> as<00:21:31.320> I<00:21:31.640> understand 00:21:31.750 --> 00:21:31.760 align:start position:0% like an improvement as I understand 00:21:31.760 --> 00:21:35.430 align:start position:0% like an improvement as I understand she's<00:21:31.919> also<00:21:32.120> stock<00:21:32.400> friendly<00:21:33.039> also<00:21:33.400> yeah 00:21:35.430 --> 00:21:35.440 align:start position:0% she's also stock friendly also yeah 00:21:35.440 --> 00:21:39.590 align:start position:0% she's also stock friendly also yeah okay<00:21:36.440> um<00:21:37.039> then<00:21:37.559> next<00:21:37.880> question<00:21:38.279> is<00:21:38.919> how<00:21:39.240> fast 00:21:39.590 --> 00:21:39.600 align:start position:0% okay um then next question is how fast 00:21:39.600 --> 00:21:42.710 align:start position:0% okay um then next question is how fast slow<00:21:40.120> is<00:21:40.360> gkr<00:21:40.880> verifier<00:21:41.440> in 00:21:42.710 --> 00:21:42.720 align:start position:0% slow is gkr verifier in 00:21:42.720 --> 00:21:45.750 align:start position:0% slow is gkr verifier in stocks<00:21:43.720> uh<00:21:43.880> yeah<00:21:44.080> good<00:21:44.279> question 00:21:45.750 --> 00:21:45.760 align:start position:0% stocks uh yeah good question 00:21:45.760 --> 00:21:49.789 align:start position:0% stocks uh yeah good question um<00:21:46.760> I<00:21:47.120> it<00:21:47.559> I<00:21:47.679> think<00:21:47.880> it's<00:21:48.600> comparable<00:21:49.240> to<00:21:49.679> uh 00:21:49.789 --> 00:21:49.799 align:start position:0% um I it I think it's comparable to uh 00:21:49.799 --> 00:21:52.029 align:start position:0% um I it I think it's comparable to uh from<00:21:50.000> what<00:21:50.120> I<00:21:50.279> recall<00:21:51.039> I<00:21:51.120> was<00:21:51.279> focusing<00:21:51.840> Less 00:21:52.029 --> 00:21:52.039 align:start position:0% from what I recall I was focusing Less 00:21:52.039 --> 00:21:54.750 align:start position:0% from what I recall I was focusing Less on<00:21:52.159> GK<00:21:52.760> fa<00:21:53.039> fire<00:21:53.200> so<00:21:53.279> I'd<00:21:53.440> have<00:21:53.559> to<00:21:53.679> go<00:21:53.880> back<00:21:54.640> uh 00:21:54.750 --> 00:21:54.760 align:start position:0% on GK fa fire so I'd have to go back uh 00:21:54.760 --> 00:21:57.390 align:start position:0% on GK fa fire so I'd have to go back uh it<00:21:54.880> was<00:21:55.080> not<00:21:55.640> drastically<00:21:56.640> faster<00:21:56.919> or<00:21:57.039> slower 00:21:57.390 --> 00:21:57.400 align:start position:0% it was not drastically faster or slower 00:21:57.400 --> 00:21:59.350 align:start position:0% it was not drastically faster or slower it<00:21:57.480> was<00:21:58.000> you<00:21:58.080> were<00:21:58.600> getting<00:21:58.799> roughly<00:21:59.080> the<00:21:59.159> same 00:21:59.350 --> 00:21:59.360 align:start position:0% it was you were getting roughly the same 00:21:59.360 --> 00:22:01.110 align:start position:0% it was you were getting roughly the same maybe<00:21:59.559> a<00:21:59.679> bit<00:21:59.799> slower<00:22:00.520> okay<00:22:00.679> and<00:22:00.840> does<00:22:00.960> it 00:22:01.110 --> 00:22:01.120 align:start position:0% maybe a bit slower okay and does it 00:22:01.120 --> 00:22:04.149 align:start position:0% maybe a bit slower okay and does it change<00:22:01.520> like<00:22:02.080> in<00:22:02.279> terms<00:22:02.600> of<00:22:03.039> the<00:22:03.240> size<00:22:03.520> of<00:22:03.960> the 00:22:04.149 --> 00:22:04.159 align:start position:0% change like in terms of the size of the 00:22:04.159 --> 00:22:07.110 align:start position:0% change like in terms of the size of the lookup<00:22:04.640> you're<00:22:04.919> proven<00:22:05.600> or<00:22:06.279> yes<00:22:06.559> but<00:22:06.799> in<00:22:06.960> the 00:22:07.110 --> 00:22:07.120 align:start position:0% lookup you're proven or yes but in the 00:22:07.120 --> 00:22:09.230 align:start position:0% lookup you're proven or yes but in the numbers<00:22:07.520> that<00:22:07.720> we<00:22:08.120> that<00:22:08.279> we<00:22:08.559> care<00:22:08.799> about<00:22:09.039> so 00:22:09.230 --> 00:22:09.240 align:start position:0% numbers that we that we care about so 00:22:09.240 --> 00:22:12.590 align:start position:0% numbers that we that we care about so maybe<00:22:09.520> traces<00:22:09.880> of<00:22:10.080> 2<00:22:10.279> to<00:22:10.440> the<00:22:10.640> 16<00:22:11.640> or<00:22:12.400> we 00:22:12.590 --> 00:22:12.600 align:start position:0% maybe traces of 2 to the 16 or we 00:22:12.600 --> 00:22:16.029 align:start position:0% maybe traces of 2 to the 16 or we typically<00:22:13.400> around<00:22:13.720> that<00:22:14.240> uh<00:22:14.799> you<00:22:15.000> didn't<00:22:15.200> see 00:22:16.029 --> 00:22:16.039 align:start position:0% typically around that uh you didn't see 00:22:16.039 --> 00:22:17.909 align:start position:0% typically around that uh you didn't see many<00:22:16.279> differences<00:22:16.840> between<00:22:17.320> uh<00:22:17.640> it<00:22:17.720> was 00:22:17.909 --> 00:22:17.919 align:start position:0% many differences between uh it was 00:22:17.919 --> 00:22:20.909 align:start position:0% many differences between uh it was relatively<00:22:18.640> constant<00:22:19.440> okay<00:22:19.679> cool 00:22:20.909 --> 00:22:20.919 align:start position:0% relatively constant okay cool 00:22:20.919 --> 00:22:23.710 align:start position:0% relatively constant okay cool cool<00:22:21.919> uh<00:22:22.120> if<00:22:22.240> we<00:22:22.400> can<00:22:22.679> just<00:22:22.840> scroll<00:22:23.279> down<00:22:23.520> to 00:22:23.710 --> 00:22:23.720 align:start position:0% cool uh if we can just scroll down to 00:22:23.720 --> 00:22:27.350 align:start position:0% cool uh if we can just scroll down to see<00:22:24.679> more 00:22:27.350 --> 00:22:27.360 align:start position:0% 00:22:27.360 --> 00:22:29.830 align:start position:0% questions<00:22:28.360> and<00:22:28.559> yeah<00:22:28.720> miden<00:22:29.159> law<00:22:29.440> yeah<00:22:29.679> good 00:22:29.830 --> 00:22:29.840 align:start position:0% questions and yeah miden law yeah good 00:22:29.840 --> 00:22:31.470 align:start position:0% questions and yeah miden law yeah good one<00:22:30.080> it's<00:22:30.200> a<00:22:30.360> good 00:22:31.470 --> 00:22:31.480 align:start position:0% one it's a good 00:22:31.480 --> 00:22:34.870 align:start position:0% one it's a good one<00:22:32.480> and<00:22:32.880> maybe<00:22:33.200> last<00:22:33.520> question<00:22:34.320> any<00:22:34.600> plans 00:22:34.870 --> 00:22:34.880 align:start position:0% one and maybe last question any plans 00:22:34.880 --> 00:22:37.430 align:start position:0% one and maybe last question any plans for<00:22:35.120> Hardware<00:22:35.600> acceleration<00:22:36.080> for<00:22:36.200> the<00:22:36.320> mid<00:22:36.679> V 00:22:37.430 --> 00:22:37.440 align:start position:0% for Hardware acceleration for the mid V 00:22:37.440 --> 00:22:41.830 align:start position:0% for Hardware acceleration for the mid V yeah<00:22:38.440> uh<00:22:38.840> so<00:22:39.080> we<00:22:39.279> have<00:22:39.799> uh<00:22:40.799> GPU<00:22:41.279> support<00:22:41.679> that 00:22:41.830 --> 00:22:41.840 align:start position:0% yeah uh so we have uh GPU support that 00:22:41.840 --> 00:22:45.510 align:start position:0% yeah uh so we have uh GPU support that is<00:22:42.279> maybe<00:22:42.720> even<00:22:43.159> in<00:22:43.480> our<00:22:44.480> main<00:22:44.760> branch<00:22:45.120> yet<00:22:45.360> or 00:22:45.510 --> 00:22:45.520 align:start position:0% is maybe even in our main branch yet or 00:22:45.520 --> 00:22:48.950 align:start position:0% is maybe even in our main branch yet or at<00:22:45.679> least<00:22:45.960> soon<00:22:46.279> to<00:22:46.440> be<00:22:47.360> uh<00:22:48.240> but<00:22:48.600> we<00:22:48.720> have 00:22:48.950 --> 00:22:48.960 align:start position:0% at least soon to be uh but we have 00:22:48.960 --> 00:22:52.710 align:start position:0% at least soon to be uh but we have active<00:22:49.520> GPU<00:22:50.000> work<00:22:50.279> being<00:22:50.640> done<00:22:51.640> uh<00:22:52.520> more 00:22:52.710 --> 00:22:52.720 align:start position:0% active GPU work being done uh more 00:22:52.720 --> 00:22:55.070 align:start position:0% active GPU work being done uh more longterm<00:22:53.120> we're<00:22:53.360> also<00:22:53.600> looking<00:22:53.919> at<00:22:54.400> uh<00:22:54.480> using 00:22:55.070 --> 00:22:55.080 align:start position:0% longterm we're also looking at uh using 00:22:55.080 --> 00:22:57.990 align:start position:0% longterm we're also looking at uh using ZK<00:22:55.919> specific<00:22:56.720> chips<00:22:57.360> but<00:22:57.480> that<00:22:57.600> are<00:22:57.760> not<00:22:57.880> on 00:22:57.990 --> 00:22:58.000 align:start position:0% ZK specific chips but that are not on 00:22:58.000 --> 00:23:01.190 align:start position:0% ZK specific chips but that are not on the<00:22:58.080> market<00:22:58.320> yet<00:22:58.799> but<00:22:59.039> it's<00:22:59.200> already<00:22:59.799> a<00:23:00.279> okay 00:23:01.190 --> 00:23:01.200 align:start position:0% the market yet but it's already a okay 00:23:01.200 --> 00:23:04.310 align:start position:0% the market yet but it's already a okay yeah<00:23:01.440> yeah<00:23:01.720> great<00:23:02.000> okay<00:23:02.120> cool<00:23:02.400> cool<00:23:03.400> well<00:23:04.159> well 00:23:04.310 --> 00:23:04.320 align:start position:0% yeah yeah great okay cool cool well well 00:23:04.320 --> 00:23:06.549 align:start position:0% yeah yeah great okay cool cool well well if<00:23:04.480> we<00:23:05.039> give<00:23:05.200> a<00:23:05.360> round<00:23:05.559> of<00:23:05.679> applause<00:23:06.039> of<00:23:06.400> for 00:23:06.549 --> 00:23:06.559 align:start position:0% if we give a round of applause of for 00:23:06.559 --> 00:23:08.830 align:start position:0% if we give a round of applause of for our<00:23:06.720> speaker<00:23:07.039> Philip<00:23:07.679> thank<00:23:07.840> you<00:23:08.000> very<00:23:08.200> much 00:23:08.830 --> 00:23:08.840 align:start position:0% our speaker Philip thank you very much 00:23:08.840 --> 00:23:12.000 align:start position:0% our speaker Philip thank you very much thank<00:23:09.000> you