fix the _trusted setup_ size

in 3c91ef12 and !54 (merged), a new implementation of the creation of the trusted setup has been introduced, that gets rid of the E: Pairing requirement with a more general <F: PrimeField, G: CurveGroup<_>>.

however, the size of the trusted setup was incorrect. zk::setup requires the maximum degree of the trusted setup, however, the number of bytes nb_bytes was consistently being given to it throughout the code base...

this MR

• introduces a new zk::nb_elements_in_setup that converts a number of bytes to the associated number of trusted setup elements
• uses that new zk function before calling zK::setup in all the code base

results

💡 Note
!58 (merged) is required for the whole table to be used easily

💡 Note
here is how to run the benchmarks in Nushell

let bad_mr = "3c91ef1"
let fix = "fix-setup-size"

git co $"($bad_mr)^"
cargo criterion --output-format verbose --message-format json out> benches/results/before.ndjson
cargo run --example bench_setup_size out>> benches/results/before.ndjson

git co $bad_mr
cargo criterion --output-format verbose --message-format json out> benches/results/after.ndjson
cargo run --example bench_setup_size out>> benches/results/after.ndjson

git co $fix
cargo criterion --output-format verbose --message-format json out> benches/results/fix.ndjson
cargo run --example bench_setup_size out>> benches/results/fix.ndjson

and here the script used to generate that table is the following:

def "parse bench-file" []: table<reason: string, id: string, mean: any> -> table<id: string, mean: float> {
    where reason == "benchmark-complete"
        | select id mean
        # NOTE: because `bench_setup_size.rs` outputs `record<reason: string, id: string, mean: float>`
        | update mean { if ($in | describe) == int { $in } else { $in.estimate } }
        # NOTE: addressed in `!58`
        | update id {|it|
            if ($it.id | str starts-with "recoding") {
                $it.id ++ " on some curve"
            } else {
                $it.id
            }
        }
        | update mean { into int }
        | update id { parse "{id} on {curve}" | into record | get id }
}

let before = open benches/results/before.ndjson | parse bench-file
let after = open benches/results/after.ndjson | parse bench-file
let fix = open benches/results/fix.ndjson | parse bench-file

$before
    | join $after id
    | rename --column { mean: "before", mean_: "after" }
    | join $fix id
    | rename --column { mean: "fix" }
    | insert b->a {|it| $it.after / $it.before | math round --precision 2 }
    | insert a->f {|it| $it.fix / $it.after | math round --precision 2 }
    | insert b->f {|it| $it.fix / $it.before | math round --precision 2 }
    | select id before b->a after a->f fix b->f
    | to md --pretty

❗ Important
before this very MR, i.e. on 3c91ef1, there was a factor of 15x between before and after, meaning that the trusted setups were 15 times larger and longer to serde 👀

this can be explained by the following facts

• due to the bad sizes given to the trusted setup building function, the setups were around 30 times larger, 30 being close to the size of a field element on BLS-12-381
• because the zk::setup function only creates half of what its Arkworks counterpart does, the setups were at the same time around 2 times smaller

combining these two and we get a factor of 15x!!

now, with this MR, we get rid of the first factor and are left with trusted setups twice as small and twice as fast to serde 🎉

id	before	b->a	after	a->f	fix	b->f
inverse 10x10	336359	0.93	313852	1.05	329191	0.98
inverse 15x15	811018	0.99	800064	1.01	807417	1
inverse 20x20	1511592	1	1508034	1.02	1542538	1.02
inverse 30x30	3703750	1.01	3731380	1.02	3793071	1.02
inverse 40x40	7163839	1	7145015	1.03	7336996	1.02
inverse 60x60	18620089	1	18625577	1.02	18922329	1.02
inverse 80x80	37571610	1	37643906	1.02	38306236	1.02
inverse 120x120	105404054	1	105281874	1.01	106797441	1.01
inverse 160x160	224332257	1	224092724	1.01	227066824	1.01
inverse 240x240	671096671	1	671005055	1.01	679280010	1.01
inverse 320x320	1487909175	1	1488534950	1.01	1506027089	1.01
transpose 10x10	87	0.93	81	1	81	0.93
transpose 15x15	175	0.96	168	1	168	0.96
transpose 20x20	284	1.03	293	0.95	279	0.98
transpose 30x30	759	1.22	924	0.89	823	1.08
transpose 40x40	1798	1.63	2935	0.98	2887	1.61
transpose 60x60	3830	1.67	6378	1.01	6468	1.69
transpose 80x80	7720	1.5	11548	0.99	11470	1.49
transpose 120x120	16365	1.5	24572	0.98	24059	1.47
transpose 160x160	42764	1.18	50453	1.07	54189	1.27
transpose 240x240	119435	1.18	141357	1	140752	1.18
transpose 320x320	218674	1.13	246262	1	247167	1.13
mul 10x10	15499	1	15474	1	15527	1
mul 15x15	51800	1	51913	1	51772	1
mul 20x20	122399	1	122390	1.01	123248	1.01
mul 30x30	499047	0.95	474740	1.01	481756	0.97
mul 40x40	1224755	0.98	1203588	1.01	1211995	0.99
mul 60x60	4166589	0.99	4122003	1	4139839	0.99
mul 80x80	9942560	0.99	9870864	1	9912815	1
mul 120x120	33706366	0.99	33458234	1.01	33680802	1
mul 160x160	79645646	1	79974020	1.01	80469214	1.01
mul 240x240	277091998	0.99	274638961	1.01	276412347	1
mul 320x320	664942845	1	662229758	1.02	676065811	1.02
recoding 1 bytes and 2 shards with k = 2	124	1	124	1.02	127	1.02
recoding 1 bytes and 2 shards with k = 4	179	0.99	178	1.01	180	1.01
recoding 1 bytes and 2 shards with k = 8	284	1	284	1	285	1
recoding 1 bytes and 2 shards with k = 16	496	1.01	499	1.01	505	1.02
recoding 1 bytes and 4 shards with k = 2	347	1.01	349	0.99	347	1
recoding 1 bytes and 4 shards with k = 4	505	1	505	1	507	1
recoding 1 bytes and 4 shards with k = 8	821	1	825	1	825	1
recoding 1 bytes and 4 shards with k = 16	1451	1	1454	1.01	1464	1.01
recoding 1 bytes and 8 shards with k = 2	792	1	791	1	792	1
recoding 1 bytes and 8 shards with k = 4	1162	1	1163	1.01	1169	1.01
recoding 1 bytes and 8 shards with k = 8	1884	1.01	1897	1	1902	1.01
recoding 1 bytes and 8 shards with k = 16	3361	1	3368	1.02	3446	1.03
recoding 1 bytes and 16 shards with k = 2	1680	1	1679	1.01	1699	1.01
recoding 1 bytes and 16 shards with k = 4	2472	1	2475	1	2468	1
recoding 1 bytes and 16 shards with k = 8	4034	1	4033	1.01	4060	1.01
recoding 1 bytes and 16 shards with k = 16	7187	1	7173	1.02	7331	1.02
recoding 1024 bytes and 2 shards with k = 2	1020	1	1020	1	1017	1
recoding 1024 bytes and 2 shards with k = 4	1079	1	1081	0.98	1064	0.99
recoding 1024 bytes and 2 shards with k = 8	1186	0.98	1167	1	1166	0.98
recoding 1024 bytes and 2 shards with k = 16	1386	1	1392	0.99	1383	1
recoding 1024 bytes and 4 shards with k = 2	2978	1	2968	1	2970	1
recoding 1024 bytes and 4 shards with k = 4	3120	1	3113	1	3113	1
recoding 1024 bytes and 4 shards with k = 8	3438	1	3445	1	3447	1
recoding 1024 bytes and 4 shards with k = 16	4056	1	4071	1	4051	1
recoding 1024 bytes and 8 shards with k = 2	6905	1	6879	1	6861	0.99
recoding 1024 bytes and 8 shards with k = 4	7236	1	7216	1	7227	1
recoding 1024 bytes and 8 shards with k = 8	7969	1	7986	1	7962	1
recoding 1024 bytes and 8 shards with k = 16	9455	1	9427	1	9442	1
recoding 1024 bytes and 16 shards with k = 2	14746	1	14760	0.99	14686	1
recoding 1024 bytes and 16 shards with k = 4	15516	1	15493	1	15538	1
recoding 1024 bytes and 16 shards with k = 8	17112	1	17097	1	17078	1
recoding 1024 bytes and 16 shards with k = 16	20237	1	20284	1	20295	1
recoding 1048576 bytes and 2 shards with k = 2	1427516	1.01	1441658	0.99	1424866	1
recoding 1048576 bytes and 2 shards with k = 4	1083761	1.01	1094451	1	1089954	1.01
recoding 1048576 bytes and 2 shards with k = 8	1087564	0.99	1076515	1.02	1094795	1.01
recoding 1048576 bytes and 2 shards with k = 16	1089556	0.99	1078406	1.03	1105840	1.01
recoding 1048576 bytes and 4 shards with k = 2	3256507	1	3250060	1.04	3370007	1.03
recoding 1048576 bytes and 4 shards with k = 4	3259079	1.01	3285892	1	3297768	1.01
recoding 1048576 bytes and 4 shards with k = 8	3235697	1	3244151	1.01	3278027	1.01
recoding 1048576 bytes and 4 shards with k = 16	3240586	1.01	3264910	1.01	3284101	1.01
recoding 1048576 bytes and 8 shards with k = 2	7580388	1	7576306	1.02	7732461	1.02
recoding 1048576 bytes and 8 shards with k = 4	7567385	1.01	7614250	1.01	7699032	1.02
recoding 1048576 bytes and 8 shards with k = 8	7589588	1	7584071	1.01	7643021	1.01
recoding 1048576 bytes and 8 shards with k = 16	7572517	1	7596138	1.01	7637596	1.01
recoding 1048576 bytes and 16 shards with k = 2	16248634	1	16245477	1.01	16450530	1.01
recoding 1048576 bytes and 16 shards with k = 4	16253850	1	16299266	1.01	16458170	1.01
recoding 1048576 bytes and 16 shards with k = 8	16240827	1	16265027	1	16256734	1
recoding 1048576 bytes and 16 shards with k = 16	16229981	1	16307729	1	16265882	1
setup/setup 1024	8934763	2.12	18942383	0.11	2175852	0.24
setup/serializing with compression 1024	4194	15.82	66364	0.03	2100	0.5
setup/serializing with no compression 1024	4953	16.04	79451	0.03	2501	0.5
setup/deserializing with compression and validation 1024	3644409	15.18	55337980	0.03	1809773	0.5
setup/deserializing with compression and no validation 1024	1065186	15.74	16762363	0.03	544255	0.51
setup/deserializing with no compression and validation 1024	2566945	15.17	38931135	0.03	1258935	0.49
setup/deserializing with no compression and no validation 1024	6722	14.84	99769	0.03	3235	0.48
setup/setup 2048	9092980	3.63	33024605	0.09	2909175	0.32
setup/serializing with compression 2048	8240	16.32	134437	0.03	4141	0.5
setup/serializing with no compression 2048	9767	16.41	160306	0.03	4976	0.51
setup/deserializing with compression and validation 2048	7239787	15.32	110931280	0.03	3639477	0.5
setup/deserializing with compression and no validation 2048	2113330	15.93	33674890	0.03	1084243	0.51
setup/deserializing with no compression and validation 2048	5081373	15.25	77482178	0.03	2537317	0.5
setup/deserializing with no compression and no validation 2048	13079	15.14	198034	0.03	6479	0.5
setup/setup 4096	9731992	6.14	59757543	0.07	4328023	0.44
setup/serializing with compression 4096	16462	16.44	270647	0.03	8407	0.51
setup/serializing with no compression 4096	19654	16.4	322264	0.03	9854	0.5
setup/deserializing with compression and validation 4096	14330104	15.47	221659652	0.03	7227388	0.5
setup/deserializing with compression and no validation 4096	4214098	15.79	66537465	0.03	2137818	0.51
setup/deserializing with no compression and validation 4096	10095359	15.33	154755178	0.03	5037809	0.5
setup/deserializing with no compression and no validation 4096	26192	14.94	391397	0.03	12862	0.49
setup/setup 8192	9594720	11.35	108884342	0.06	6893620	0.72
setup/serializing with compression 8192	33114	16.42	543855	0.03	16713	0.5
setup/serializing with no compression 8192	39992	16.17	646576	0.03	19983	0.5
setup/deserializing with compression and validation 8192	28578044	15.55	444525236	0.03	14337421	0.5
setup/deserializing with compression and no validation 8192	8417684	15.93	134082205	0.03	4309633	0.51
setup/deserializing with no compression and validation 8192	20134851	15.39	309785238	0.03	10066797	0.5
setup/deserializing with no compression and no validation 8192	51832	15.06	780369	0.03	25710	0.5
setup/setup 16384	10096523	19.72	199105054	0.06	11317161	1.12
setup/serializing with compression 16384	67050	16.28	1091282	0.03	33502	0.5
setup/serializing with no compression 16384	80269	16.2	1300111	0.03	40785	0.51
setup/deserializing with compression and validation 16384	56905556	15.56	885542593	0.03	28622218	0.5
setup/deserializing with compression and no validation 16384	16829951	15.96	268660355	0.03	8607645	0.51
setup/deserializing with no compression and validation 16384	40158772	15.44	619890738	0.03	20006634	0.5
setup/deserializing with no compression and no validation 16384	103242	15.07	1555913	0.03	51533	0.5
serialized size with compression and validation 1024	3280	15	49208	0.03	1640	0.5
serialized size with compression and no validation 1024	3280	15	49208	0.03	1640	0.5
serialized size with no compression and validation 1024	6544	15.04	98408	0.03	3272	0.5
serialized size with no compression and no validation 1024	6544	15.04	98408	0.03	3272	0.5
serialized size with compression and validation 2048	6448	15.25	98360	0.03	3224	0.5
serialized size with compression and no validation 2048	6448	15.25	98360	0.03	3224	0.5
serialized size with no compression and validation 2048	12880	15.27	196712	0.03	6440	0.5
serialized size with no compression and no validation 2048	12880	15.27	196712	0.03	6440	0.5
serialized size with compression and validation 4096	12784	15.38	196664	0.03	6392	0.5
serialized size with compression and no validation 4096	12784	15.38	196664	0.03	6392	0.5
serialized size with no compression and validation 4096	25552	15.39	393320	0.03	12776	0.5
serialized size with no compression and no validation 4096	25552	15.39	393320	0.03	12776	0.5
serialized size with compression and validation 8192	25456	15.45	393272	0.03	12728	0.5
serialized size with compression and no validation 8192	25456	15.45	393272	0.03	12728	0.5
serialized size with no compression and validation 8192	50896	15.45	786536	0.03	25448	0.5
serialized size with no compression and no validation 8192	50896	15.45	786536	0.03	25448	0.5
serialized size with compression and validation 16384	50800	15.48	786488	0.03	25400	0.5
serialized size with compression and no validation 16384	50800	15.48	786488	0.03	25400	0.5
serialized size with no compression and validation 16384	101584	15.48	1572968	0.03	50792	0.5
serialized size with no compression and no validation 16384	101584	15.48	1572968	0.03	50792	0.5