diff --git a/noir-projects/noir-protocol-circuits/Nargo.template.toml b/noir-projects/noir-protocol-circuits/Nargo.template.toml index b2f991f7e9a..0ff8fc258ef 100644 --- a/noir-projects/noir-protocol-circuits/Nargo.template.toml +++ b/noir-projects/noir-protocol-circuits/Nargo.template.toml @@ -1,6 +1,7 @@ [workspace] members = [ "crates/types", + "crates/blob", "crates/parity-base", "crates/parity-lib", "crates/parity-root", diff --git a/noir-projects/noir-protocol-circuits/crates/blob/Nargo.toml b/noir-projects/noir-protocol-circuits/crates/blob/Nargo.toml new file mode 100644 index 00000000000..208a356b56f --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/Nargo.toml @@ -0,0 +1,8 @@ +[package] +name = "blob" +type = "bin" +authors = [""] +compiler_version = ">=0.30.0" + +[dependencies] +bigint = {tag = "v0.3.0", git = "https://github.com/noir-lang/noir-bignum" } \ No newline at end of file diff --git a/noir-projects/noir-protocol-circuits/crates/blob/README.md b/noir-projects/noir-protocol-circuits/crates/blob/README.md new file mode 100644 index 00000000000..7dfb6521162 --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/README.md @@ -0,0 +1,24 @@ +## Constraint counts + +Current count: 428,240. + +## Compile: + +`time nargo compile` + +## Get gate count: + +Install bbup: `curl -L https://raw.githubusercontent.com/AztecProtocol/aztec-packages/master/barretenberg/cpp/installation/install | bash` +The version of bb compatible with nargo 0.33.0 is `bbup --version 0.48.0` + +`time bb gates_mega_honk -b ./target/blob.json` + +## Generate, then serve a new flamegraph, after compiling: + + + +`~/packages/noir/noir-repo/target/release/noir-profiler gates-flamegraph --artifact-path ./target/blob.json --backend-path ~/.bb/bb --output ./flamegraph --backend-gates-command "gates_mega_honk" -- -h && python3 -m http.server --directory "./flamegraph" 3000` + +## To serve an existing flamegraph: + +`python3 -m http.server --directory "./flamegraph" 3000` diff --git a/noir-projects/noir-protocol-circuits/crates/blob/flamegraph/main_gates.svg b/noir-projects/noir-protocol-circuits/crates/blob/flamegraph/main_gates.svg new file mode 100644 index 00000000000..d9e12afa971 --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/flamegraph/main_gates.svg @@ -0,0 +1,595 @@ +./target/blob.json-main Reset ZoomSearch acir::arithmetic (6 gates, 0.00%)main.nr:235:35::_kzg_commitment(kzg_commitment); + (83 gates, 0.02%)main.nr:230:20::idon2::Poseidon2::hash(kzg_commitment, kzg_commitment.len()); + (83 gates, 0.02%)poseidon2.nr:17:13::Poseidon2::hash_internal(input, N, false) (83 gates, 0.02%)poseidon2.nr:114:9::sponge.squeeze() (83 gates, 0.02%)poseidon2.nr:79:39::self.perform_duplex() (83 gates, 0.02%)poseidon2.nr:40:22::crate::hash::poseidon2_permutation(self.state, 4) (83 gates, 0.02%)acir::blackbox::poseidon2_permutation (77 gates, 0.02%)main.nr:224:34::_to_fields__tightly_packed(blob); + (4,064 gates, 0.94%)main.nr:197:69::fe_extract_top_bit(blob[i]); + (4,064 gates, 0.94%)main.nr:180:9::rt_eq(top_bit * 0x8000 + top_limb_with_top_bit_removed, x.limbs[2]); + + (4,064 gates, 0.94%)acir::arithmetic (4,064 gates, 0.94%)acir::arithmetic (9,968 gates, 2.30%)a..poseidon2.nr:104:17::sponge.absorb(input[i]) (115,458 gates, 26.64%)poseidon2.nr:104:17::sponge.absorb(input[i])poseidon2.nr:52:21::self.perform_duplex() (115,458 gates, 26.64%)poseidon2.nr:52:21::self.perform_duplex()poseidon2.nr:40:22::crate::hash::poseidon2_permutation(self.state, 4) (115,458 gates, 26.64%)poseidon2.nr:40:22::crate::hash::poseidon2_..acir::blackbox::poseidon2_permutation (105,490 gates, 24.34%)acir::blackbox::poseidon2_permutationacir::arithmetic (254 gates, 0.06%)main.nr:236:25::_blob(blob); + (119,853 gates, 27.66%)main.nr:236:25::_blob(blob); + main.nr:225:20::idon2::Poseidon2::hash(blob_as_fields, blob_as_fields.len()); + (115,789 gates, 26.72%)main.nr:225:20::idon2::Poseidon2::hash(blob..poseidon2.nr:17:13::Poseidon2::hash_internal(input, N, false) (115,789 gates, 26.72%)poseidon2.nr:17:13::Poseidon2::hash_interna..poseidon2.nr:114:9::sponge.squeeze() (331 gates, 0.08%)poseidon2.nr:79:39::self.perform_duplex() (331 gates, 0.08%)poseidon2.nr:40:22::crate::hash::poseidon2_permutation(self.state, 4) (331 gates, 0.08%)acir::blackbox::poseidon2_permutation (77 gates, 0.02%)main.nr:249:34::ute_challenge(blob, kzg_commitment); + (120,013 gates, 27.69%)main.nr:249:34::ute_challenge(blob, kzg_comm..main.nr:238:25::idon2::Poseidon2::hash([blob_hash, kzg_commitment_hash], 2); + (77 gates, 0.02%)poseidon2.nr:17:13::Poseidon2::hash_internal(input, N, false) (77 gates, 0.02%)poseidon2.nr:114:9::sponge.squeeze() (77 gates, 0.02%)poseidon2.nr:79:39::self.perform_duplex() (77 gates, 0.02%)poseidon2.nr:40:22::crate::hash::poseidon2_permutation(self.state, 4) (77 gates, 0.02%)acir::blackbox::poseidon2_permutation (77 gates, 0.02%)main.nr:250:40::d_to_bignum(challenge_z); + + (1 gates, 0.00%)main.nr:108:16::k == x); + + (1 gates, 0.00%)acir::arithmetic (1 gates, 0.00%)runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (2,789 gates, 0.64%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (2,789 gates, 0.64%)acir::blackbox::range (2,789 gates, 0.64%)runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (2,814 gates, 0.65%)runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (25 gates, 0.01%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (25 gates, 0.01%)acir::blackbox::range (25 gates, 0.01%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (1 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (1 gates, 0.00%)acir::blackbox::range (1 gates, 0.00%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (3 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (3 gates, 0.00%)acir::blackbox::range (3 gates, 0.00%)acir::arithmetic (17 gates, 0.00%)runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (38 gates, 0.01%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (38 gates, 0.01%)acir::blackbox::range (21 gates, 0.00%)main.nr:290:13::evaluate_quadratic_expression([[z]], [[false]], [[z]], [[false]], [t1], [true]); + + l (2,858 gates, 0.66%)lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (2,858 gates, 0.66%)runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (2 gates, 0.00%)acir::arithmetic (2 gates, 0.00%)runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (115 gates, 0.03%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (115 gates, 0.03%)acir::blackbox::range (115 gates, 0.03%)runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (131 gates, 0.03%)runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (16 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (16 gates, 0.00%)acir::blackbox::range (16 gates, 0.00%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (11 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (11 gates, 0.00%)acir::blackbox::range (11 gates, 0.00%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (33 gates, 0.01%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (33 gates, 0.01%)acir::blackbox::range (33 gates, 0.01%)acir::arithmetic (187 gates, 0.04%)runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (418 gates, 0.10%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (418 gates, 0.10%)acir::blackbox::range (231 gates, 0.05%)main.nr:301:17::evaluate_quadratic_expression([[t1]], [[false]], [[t1]], [[false]], [t2], [true]); + + (615 gates, 0.14%)lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (615 gates, 0.14%)runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (22 gates, 0.01%)acir::arithmetic (22 gates, 0.01%)runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (11 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (11 gates, 0.00%)acir::blackbox::range (11 gates, 0.00%)runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (13 gates, 0.00%)runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (2 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (2 gates, 0.00%)acir::blackbox::range (2 gates, 0.00%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (1 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (1 gates, 0.00%)acir::blackbox::range (1 gates, 0.00%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (3 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (3 gates, 0.00%)acir::blackbox::range (3 gates, 0.00%)acir::arithmetic (14 gates, 0.00%)runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (35 gates, 0.01%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (35 gates, 0.01%)acir::blackbox::range (21 gates, 0.00%)main.nr:324:13::evaluate_quadratic_expression( + [[z_pow_d]], + [[false]], + [[D_INV]], + [[false]], + [factor, D_INV], + [true, true] + ); + + / (53 gates, 0.01%)lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (53 gates, 0.01%)runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (1 gates, 0.00%)acir::arithmetic (1 gates, 0.00%)runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (43,008 gates, 9.92%)runtime_bignum..mod.nr:27:9::self.__assert_max_bit_size(bit_size) (43,008 gates, 9.92%)mod.nr:27:9::s..acir::blackbox::range (43,008 gates, 9.92%)acir::blackbox..runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (49,152 gates, 11.34%)runtime_bignum.nr..runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (6,144 gates, 1.42%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (6,144 gates, 1.42%)acir::blackbox::range (6,144 gates, 1.42%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (4,096 gates, 0.95%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (4,096 gates, 0.95%)acir::blackbox::range (4,096 gates, 0.95%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (12,288 gates, 2.84%)ru..mod.nr:27:9::self.__assert_max_bit_size(bit_size) (12,288 gates, 2.84%)mo..acir::blackbox::range (12,288 gates, 2.84%)ac..acir::arithmetic (102,397 gates, 23.63%)acir::arithmeticruntime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (188,413 gates, 43.48%)runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (188,413 gates, 43.48%)mod.nr:27:9::self.__assert_max_bit_size(bit_size)acir::blackbox::range (86,016 gates, 19.85%)acir::blackbox::rangemain.nr:403:29::dratic_expression( + [[fracs[i]]], + [[false]], + [[z, NEGATIVE_ROOTS[i]]], + [[false, false]], + [ys[i]], + [true] + ); + } + + // OK s (266,235 gates, 61.44%)main.nr:403:29::dratic_expression( + [[fracs[i]]], + [[false]], + [[z, N..lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (266,235 gates, 61.44%)lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + ..runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (12,286 gates, 2.84%)ru..acir::arithmetic (12,286 gates, 2.84%)ac..runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (5,376 gates, 1.24%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (5,376 gates, 1.24%)acir::blackbox::range (5,376 gates, 1.24%)runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (6,144 gates, 1.42%)runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (768 gates, 0.18%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (768 gates, 0.18%)acir::blackbox::range (768 gates, 0.18%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (512 gates, 0.12%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (512 gates, 0.12%)acir::blackbox::range (512 gates, 0.12%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (1,536 gates, 0.35%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (1,536 gates, 0.35%)acir::blackbox::range (1,536 gates, 0.35%)acir::arithmetic (22,011 gates, 5.08%)acir::..runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (32,763 gates, 7.56%)runtime_bi..mod.nr:27:9::self.__assert_max_bit_size(bit_size) (32,763 gates, 7.56%)mod.nr:27:..acir::blackbox::range (10,752 gates, 2.48%)ac..main.nr:494:45::n( + [[lhs[0]], [lhs[1]], [lhs[2]], [lhs[3]], [lhs[4]], [lhs[5]], [lhs[6]], [lhs[7]]], + [[false], [false], [false], [false], [false], [false], [false], [false]], + [[rhs[0]], [rhs[1]], [rhs[2]], [rhs[3]], [rhs[4]], [rhs[5]], [rhs[6]], [rhs[7]]], + [[false], [false], [false], [false], [false], [false], [false], [false]], + [sum, sum_out], + [false, true] + ); + + sum = sum_out; + st (43,514 gates, 10.04%)main.nr:494:45:..lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (43,514 gates, 10.04%)lib.nr:230:9::P..runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (2,559 gates, 0.59%)acir::arithmetic (2,559 gates, 0.59%)runtime_bignum.nr:251:13::self.limbs[i].assert_max_bit_size(120) (10 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (10 gates, 0.00%)acir::blackbox::range (10 gates, 0.00%)runtime_bignum.nr:420:9::quotient.validate_quotient_in_range() (12 gates, 0.00%)runtime_bignum.nr:255:9::self.limbs[N - 1].assert_max_bit_size(final_limb_bits as u32) (2 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (2 gates, 0.00%)acir::blackbox::range (2 gates, 0.00%)runtime_bignum.nr:504:9::borrow_flags.get(0).assert_max_bit_size(1) (1 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (1 gates, 0.00%)acir::blackbox::range (1 gates, 0.00%)runtime_bignum.nr:507:13::borrow_flags.get(i).assert_max_bit_size(1) (3 gates, 0.00%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (3 gates, 0.00%)acir::blackbox::range (3 gates, 0.00%)acir::arithmetic (22 gates, 0.01%)runtime_bignum.nr:531:13::product_limbs.get(i).assert_max_bit_size(126) (43 gates, 0.01%)mod.nr:27:9::self.__assert_max_bit_size(bit_size) (43 gates, 0.01%)acir::blackbox::range (21 gates, 0.00%)main.nr:252:20::centric_evaluate_blob_at_z(challenge_z_as_bignum, blob); + (313,336 gates, 72.30%)main.nr:252:20::centric_evaluate_blob_at_z(challenge_z_as_bignum, blob); + main.nr:515:41::n([[factor]], [[false]], [[sum]], [[false]], [y], [true]); + + println(y); + + y +} + +// Does (61 gates, 0.01%)lib.nr:230:9::Params::get_instance().evaluate_quadratic_expression( + lhs_terms, + lhs_flags, + rhs_terms, + rhs_flags, + linear_terms, + linear_flags + ) (61 gates, 0.01%)runtime_bignum.nr:535:16::product_limbs.get(N + N - 2) == 0 (2 gates, 0.00%)acir::arithmetic (2 gates, 0.00%)all (433,356 gates, 100%)unknown (6 gates, 0.00%)acir::arithmetic (6 gates, 0.00%) \ No newline at end of file diff --git a/noir-projects/noir-protocol-circuits/crates/blob/src/config.nr b/noir-projects/noir-protocol-circuits/crates/blob/src/config.nr new file mode 100644 index 00000000000..5062a5506a3 --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/src/config.nr @@ -0,0 +1,4110 @@ +use dep::bigint::{BigNum, fields::bls12_381Fr::BLS12_381_Fr_Params}; + +type F = BigNum<3, BLS12_381_Fr_Params>; + +global FIELDS_PER_BLOB: u64 = 4096; +global LOG_FIELDS_PER_BLOB: u64 = 12; +global EXTRA_FIELDS_PER_BLOB: u64 = 16; // 16 = floor(4096 FIELDS_PER_BLOB / 254 noir_field_bits), wasting only 32 bits. +global NOIR_FIELDS_PER_BLOB: u64 = FIELDS_PER_BLOB + EXTRA_FIELDS_PER_BLOB; +global FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB: u64 = EXTRA_FIELDS_PER_BLOB * 254; // EXTRA_FIELDS_PER_BLOB * 254 = 4064. So the first 4064 bls Fr fields in the blob will carry an extra bit in their 255th bit position, that will be used to reconstitute 16 extra fields. +global D: F = BigNum { limbs: [4096, 0, 0] }; +global D_INV = BigNum { limbs: [0x686828bfce5c19400fffff00100001, 0x6878b46ae3705eb6a46a89213de7d3, 0x73e6] }; + +global ROOTS: [F; FIELDS_PER_BLOB] = [ + BigNum { limbs: [ 0x000000000000000000000000000001, 0x000000000000000000000000000000, 0x000000 ] } , + BigNum { limbs: [ 0xbda402fffe5bfeffffffff00000000, 0xa753299d7d483339d80809a1d80553, 0x0073ed ] } , + BigNum { limbs: [ 0x030002760300000001000000000000, 0x0000000000008d51ccce760304d0ec, 0x000000 ] } , + BigNum { limbs: [ 0xbaa40089fb5bfefffeffff00000001, 0xa753299d7d47a5e80b39939ed33467, 0x0073ed ] } , + BigNum { limbs: [ 0x8b21c28713b7007228fd3397743f7a, 0x66f603fa66e78c0625cd70d77ce2b3, 0x003457 ] } , + BigNum { limbs: [ 0x32824078eaa4fe8dd702cb688bc087, 0x405d25a31660a733b23a98ca5b22a0, 0x003f96 ] } , + BigNum { limbs: [ 0x74903694b04fd86037fe81ae99502e, 0xb22e5ce11044babc5affca86bf658e, 0x001333 ] } , + BigNum { limbs: [ 0x4913cc6b4e0c269fc8017d5166afd3, 0xf524ccbc6d03787d7d083f1b189fc5, 0x0060b9 ] } , + BigNum { limbs: [ 0xbeb312f20b6f7653ea61d87742bcce, 0xce9140267af9dd1c0af834cec32c17, 0x0020b1 ] } , + BigNum { limbs: [ 0xfef0f00df2ec88ac159e2688bd4333, 0xd8c1e977024e561dcd0fd4d314d93b, 0x00533b ] } , + BigNum { limbs: [ 0xf4e672ebc1e1bb95df4b360411fe73, 0x596e753e4fcc6e92a9c460afca4a1e, 0x004f2c ] } , + BigNum { limbs: [ 0xc8bd90143c7a436a20b4c8fbee018e, 0x4de4b45f2d7bc4a72e43a8f20dbb34, 0x0024c1 ] } , + BigNum { limbs: [ 0x7a6b6cfb0faca4807b811a823f728d, 0x919ec91f38ac5ccd4631f16edba496, 0x001edc ] } , + BigNum { limbs: [ 0x43389604eeaf5a7f847ee47dc08d74, 0x15b4607e449bd66c91d61832fc60bd, 0x005511 ] } , + BigNum { limbs: [ 0xbc96af334c36bca1abb31fb37786b9, 0xf2dd7e0c63fccabf643eda8951f257, 0x0038c7 ] } , + BigNum { limbs: [ 0x010d53ccb225425e544cdf4c887948, 0xb475ab91194b687a73c92f188612fc, 0x003b25 ] } , + BigNum { limbs: [ 0x6e88fb4c38fb8a360c60997369df4e, 0x903a157988bab4bcd40e22f55448bf, 0x0050e0 ] } , + BigNum { limbs: [ 0x4f1b07b3c56074c9f39f658c9620b3, 0x17191423f48d7e7d03f9e6ac83bc94, 0x00230d ] } , + BigNum { limbs: [ 0xbc7f62d13a6e1c3ec50c9031a36ca3, 0xc5837cb5fca206050b5832d1099726, 0x0065f6 ] } , + BigNum { limbs: [ 0x0124a02ec3ede2c13af36ece5c935e, 0xe1cface780a62d34ccafd6d0ce6e2d, 0x000df6 ] } , + BigNum { limbs: [ 0x3dc46688b5e11768cc0c58459f155b, 0x0457c83a7d9c5aea51f540eb0c0496, 0x002c7e ] } , + BigNum { limbs: [ 0x7fdf9c77487ae79733f3a6ba60eaa6, 0xa2fb6162ffabd84f8612c8b6cc00bd, 0x00476f ] } , + BigNum { limbs: [ 0xe943612401899720d4ed194fccfeb9, 0xda18a9d30564a8f0cfd2438f018c01, 0x005303 ] } , + BigNum { limbs: [ 0xd460a1dbfcd267df2b12e5b0330148, 0xcd3a7fca77e38a490835c612d67951, 0x0020e9 ] } , + BigNum { limbs: [ 0x317ae6451bb89de69679532ae1234c, 0x237e58fcced486fa69d8e4e48506e3, 0x000461 ] } , + BigNum { limbs: [ 0x8c291cbae2a361196986abd51edcb5, 0x83d4d0a0ae73ac3f6e2f24bd52fe70, 0x006f8c ] } , + BigNum { limbs: [ 0x593d6ff6dab086ee5bcecc4e7773cb, 0xb16caf96816fa3a95d2d4016e2bd45, 0x00047c ] } , + BigNum { limbs: [ 0x6466930923ab7811a43132b1888c36, 0xf5e67a06fbd88f907adac98af5480e, 0x006f70 ] } , + BigNum { limbs: [ 0xa6d195014b641082e68bc0bc50a88f, 0x5bb8ed54ae00468b04010fa5c79f62, 0x0056f3 ] } , + BigNum { limbs: [ 0x16d26dfeb2f7ee7d19743e43af5772, 0x4b9a3c48cf47ecaed406f9fc1065f1, 0x001cfa ] } , + BigNum { limbs: [ 0x452d43f6d5756f51cb57e0e3035d15, 0xb9c6e6797777851425ea12dcacdae7, 0x001579 ] } , + BigNum { limbs: [ 0x7876bf0928e68fae34a81e1cfca2ec, 0xed8c432405d0ae25b21df6c52b2a6c, 0x005e73 ] } , + BigNum { limbs: [ 0x967f4be2f951558140d032f0a9ee53, 0x6345ec055e4d14a1e27164d8fdbd2d, 0x0045af ] } , + BigNum { limbs: [ 0x2724b71d050aa97ebf2fcc0f5611ae, 0x440d3d981efb1e97f596a4c8da4826, 0x002e3e ] } , + BigNum { limbs: [ 0xf20a6f5e1709899ddf46bac40ac8e4, 0x300e9079af0b916f129332ba2dfc0b, 0x0028eb ] } , + BigNum { limbs: [ 0xcb9993a1e752756220b9443bf5371d, 0x77449923ce3ca1cac574d6e7aa0947, 0x004b02 ] } , + BigNum { limbs: [ 0xdc0ae311f00af48469ef4d246b6883, 0xad6a79b61c1a71d544f7800a7e4ae4, 0x005391 ] } , + BigNum { limbs: [ 0xe1991fee0e510a7b9610b1db94977e, 0xf9e8afe7612dc1649310899759ba6e, 0x00205b ] } , + BigNum { limbs: [ 0x2e4312e6011bf5d941e9338fb466f7, 0x0520cdfb5d9d6c54cb86cdf73e9123, 0x0044ed ] } , + BigNum { limbs: [ 0x8f60f019fd400926be16cb704b990a, 0xa2325ba21faac6e50c813baa997430, 0x002f00 ] } , + BigNum { limbs: [ 0x12abf7f3a89e7acf065a270f3c324f, 0x64fb4536c4fcf6ad66524f0376d9e4, 0x0054fa ] } , + BigNum { limbs: [ 0xaaf80b0c55bd8430f9a5d7f0c3cdb2, 0x4257e466b84b3c8c71b5ba9e612b6f, 0x001ef3 ] } , + BigNum { limbs: [ 0x2a70a615d0b8e4d2fc5e69ac5db47f, 0x40ac57f86f5e293b1d67bc8de5d9a1, 0x000e48 ] } , + BigNum { limbs: [ 0x93335cea2da31a2d03a19553a24b82, 0x66a6d1a50dea09febaa04d13f22bb2, 0x0065a5 ] } , + BigNum { limbs: [ 0x1f590ef73ba2bdc0f1357a508e5e7b, 0x00aba73798bfaf59d0fc7261da7291, 0x0058c4 ] } , + BigNum { limbs: [ 0x9e4af408c2b9413f0eca84af71a186, 0xa6a78265e48883e0070b973ffd92c2, 0x001b29 ] } , + BigNum { limbs: [ 0x02e4e461e72e18ddc3b03ea91bc267, 0x785206b5761a878d670fcb570ab3b8, 0x006358 ] } , + BigNum { limbs: [ 0xbabf1e9e172de6223c4fc056e43d9a, 0x2f0122e8072dabac70f83e4acd519b, 0x001095 ] } , + BigNum { limbs: [ 0xc3ebf43c92a949a4593e1acca2cb6c, 0x8adc7bff16bae3ee1645113940cf46, 0x0053c7 ] } , + BigNum { limbs: [ 0xf9b80ec36bb2b55ba6c1e4335d3495, 0x1c76ad9e668d4f4bc1c2f86897360c, 0x002026 ] } , + BigNum { limbs: [ 0xe74ba2b75ca477f44e14739932aa33, 0x508a14adf95959d7d47f20aa9f0259, 0x0037d3 ] } , + BigNum { limbs: [ 0xd6586048a1b7870bb1eb8b66cd55ce, 0x56c914ef83eed9620388e8f73902f9, 0x003c1a ] } , + BigNum { limbs: [ 0x9f1f01e2bbf0ac476e05bf67d4973c, 0xcc64ae610371dcd9ce528178852eaf, 0x005a50 ] } , + BigNum { limbs: [ 0x1e85011d426b52b891fa3f982b68c5, 0xdaee7b3c79d6566009b5882952d6a4, 0x00199c ] } , + BigNum { limbs: [ 0x1ae51df978cc3878f4ee1de45ab2f2, 0x03824bef73c976407b9926e20836d2, 0x006e57 ] } , + BigNum { limbs: [ 0xa2bee506858fc6870b11e11ba54d0f, 0xa3d0ddae097ebcf95c6ee2bfcfce81, 0x000596 ] } , + BigNum { limbs: [ 0x7493f160ce4cb729b4cb21179cfb0e, 0x79931cfdd8947f799cf20f675fde6a, 0x00036b ] } , + BigNum { limbs: [ 0x4910119f300f47d64b34dde86304f3, 0x2dc00c9fa4b3b3c03b15fa3a7826e9, 0x007082 ] } , + BigNum { limbs: [ 0x0664a566a603f98c15c05b1901cef2, 0xed2ec80a4115f20c57f6d7dc953305, 0x000afc ] } , + BigNum { limbs: [ 0xb73f5d9958580573ea3fa3e6fe310f, 0xba2461933c32412d801131c542d24e, 0x0068f0 ] } , + BigNum { limbs: [ 0xe20d3cfc83311c0727b36db1974ef4, 0xd5fd4e2f04c5e7caaba64af676214e, 0x0028c6 ] } , + BigNum { limbs: [ 0xdb96c6037b2ae2f8d84c914e68b10d, 0xd155db6e78824b6f2c61beab61e404, 0x004b26 ] } , + BigNum { limbs: [ 0x36750f231bcd8672d73ebbe97445d5, 0x9ddec7fa8e98e4b5243a8bda7ca378, 0x000fe0 ] } , + BigNum { limbs: [ 0x872ef3dce28e788d28c143168bba2c, 0x097461a2eeaf4e84b3cd7dc75b61db, 0x00640d ] } , + BigNum { limbs: [ 0xd0caac87f5713c5130c2c1660125be, 0x111413588742b7c68b4d7fdd60d098, 0x006898 ] } , + BigNum { limbs: [ 0xecd9567808eac2aecf3d3d99feda43, 0x963f1644f6057b734cba89c47734ba, 0x000b55 ] } , + BigNum { limbs: [ 0x30a34e5e4c7a31a0927a327c751043, 0x983de0110e23413ff88848100458b8, 0x0030d2 ] } , + BigNum { limbs: [ 0x8d00b4a1b1e1cd5f6d85cc838aefbe, 0x0f15498c6f24f1f9df7fc191d3ac9b, 0x00431b ] } , + BigNum { limbs: [ 0x1e0777f7ef73e32ef7664cb2440ed4, 0xf5b7cb3dee5f01fe51c5b744878f5d, 0x00510d ] } , + BigNum { limbs: [ 0x9f9c8b080ee81bd10899b24dbbf12d, 0xb19b5e5f8ee9313b8642525d5075f6, 0x0022df ] } , + BigNum { limbs: [ 0x608393655b6d323eae9752b92f9726, 0x1a173d90ba01c42ee040e5579a63af, 0x0015ae ] } , + BigNum { limbs: [ 0x5d206f9aa2eeccc15168ac46d068db, 0x8d3bec0cc3466f0af7c7244a3da1a4, 0x005e3f ] } , + BigNum { limbs: [ 0x136819ab9c98a27528d588439e5b12, 0x3ae1cc22ee66ae3320a20450eb9bbf, 0x001a8f ] } , + BigNum { limbs: [ 0xaa3be95461c35c8ad72a76bc61a4ef, 0x6c715d7a8ee18506b7660550ec6994, 0x00595e ] } , + BigNum { limbs: [ 0xb0deac619bda2d4000b0b3767c9928, 0x0b00588ed2cf8b98c6ffcd682aa219, 0x0035c6 ] } , + BigNum { limbs: [ 0x0cc5569e6281d1bfff4f4b898366d9, 0x9c52d10eaa78a7a111083c39ad633a, 0x003e27 ] } , + BigNum { limbs: [ 0x95392351a789a318d303a266992f63, 0x6bb22f520df225302664820cb7fbf4, 0x00058e ] } , + BigNum { limbs: [ 0x286adfae56d25be72cfc5c9966d09e, 0x3ba0fa4b6f560e09b1a3879520095f, 0x006e5f ] } , + BigNum { limbs: [ 0x47df05ce549e034eb4bb5cc301906f, 0xace8a4aeb40b597e225a1da4156c04, 0x007144 ] } , + BigNum { limbs: [ 0x75c4fd31a9bdfbb14b44a23cfe6f92, 0xfa6a84eec93cd9bbb5adebfdc2994f, 0x0002a8 ] } , + BigNum { limbs: [ 0x7d9fd5726f6b40ed9fa5c032ec0a23, 0xa72e47136966a4512dec6b0a03377e, 0x005f2f ] } , + BigNum { limbs: [ 0x40042d8d8ef0be12605a3ecd13f5de, 0x0024e28a13e18ee8aa1b9e97d4cdd5, 0x0014be ] } , + BigNum { limbs: [ 0x3accee4e32febaa4f1f87d90884c5a, 0x5f6ba59c3994a84b028e307c93637f, 0x002254 ] } , + BigNum { limbs: [ 0x82d714b1cb5d445b0e07816f77b3a7, 0x47e7840143b38aeed579d92544a1d4, 0x005199 ] } , + BigNum { limbs: [ 0x9fa49ec41505c948af25b254d71df9, 0xdf508052e8d9a67bb3d018ed258f03, 0x006c1c ] } , + BigNum { limbs: [ 0x1dff643be95635b750da4cab28e208, 0xc802a94a946e8cbe2437f0b4b27650, 0x0007d0 ] } , + BigNum { limbs: [ 0x0af27f081250e5bddd6dacba0b1d06, 0x9b0b4fa4971529805a2b04c50efc33, 0x005b75 ] } , + BigNum { limbs: [ 0xb2b183f7ec0b194222925245f4e2fb, 0x0c47d9f8e63309b97ddd04dcc90920, 0x001878 ] } , + BigNum { limbs: [ 0xffeee172f8ac74095e328eda2f8e5c, 0x8571ac60a4ce0468b4bb2b446edda9, 0x002904 ] } , + BigNum { limbs: [ 0xbdb5218d05af8af6a1cd7025d071a5, 0x21e17d3cd87a2ed1234cde5d6927a9, 0x004ae9 ] } , + BigNum { limbs: [ 0xfd4ee83f1b2499dbe21753938adf74, 0x09b8f9797bc1d44db4c7e28c3f15cb, 0x001aa9 ] } , + BigNum { limbs: [ 0xc0551ac0e33765241de8ab6c75208d, 0x9d9a302401865eec2340271598ef87, 0x005944 ] } , + BigNum { limbs: [ 0xfb82877b46d20d3f0ffba2ea149337, 0x83626bca158c398677e14a8cb7e715, 0x005815 ] } , + BigNum { limbs: [ 0xc2217b84b789f1c0f0045c15eb6cca, 0x23f0bdd367bbf9b36026bf15201e3d, 0x001bd8 ] } , + BigNum { limbs: [ 0xf0a42ec0e52a48c009cf2b0763b3d5, 0x72b23615cf703897e487b9c664354a, 0x0054b5 ] } , + BigNum { limbs: [ 0xccffd43f1931b63ff630d3f89c4c2c, 0x34a0f387add7faa1f3804fdb73d008, 0x001f38 ] } , + BigNum { limbs: [ 0xbdbdcad82928c864e8c456229dfdef, 0xd2940967b1a207e780cc7312dd5371, 0x0067ad ] } , + BigNum { limbs: [ 0xffe63827d533369b173ba8dd620212, 0xd4bf2035cba62b52573b968efab1e1, 0x000c3f ] } , + BigNum { limbs: [ 0x4b0f40448ca04c44568101d845f71f, 0x150315161c3e6e8ba6f2374a655d9d, 0x005318 ] } , + BigNum { limbs: [ 0x7294c2bb71bbb2bba97efd27ba08e2, 0x925014876109c4ae3115d25772a7b6, 0x0020d5 ] } , + BigNum { limbs: [ 0xe21a225b58e59f1d3e494f87fd6d91, 0x8bf565feb0f7bdf7f8f0d267d1ae2a, 0x00249a ] } , + BigNum { limbs: [ 0xdb89e0a4a5765fe2c1b6af78029270, 0x1b5dc39ecc507541df17373a065728, 0x004f53 ] } , + BigNum { limbs: [ 0x94a89651aca0ff810d9cff570736e9, 0x46355ac0cbecc13494e5cc0ae758c6, 0x006c43 ] } , + BigNum { limbs: [ 0x28fb6cae51baff7ef262ffa8f8c918, 0x611dcedcb15b720543223d96f0ac8d, 0x0007aa ] } , + BigNum { limbs: [ 0x9486bb67912a2b440dab45cd1fac5b, 0x03d7743e9a0675583bdb136e9f86e1, 0x001a9d ] } , + BigNum { limbs: [ 0x291d47986d31d3bbf254b932e053a6, 0xa37bb55ee341bde19c2cf633387e72, 0x005950 ] } , + BigNum { limbs: [ 0x6fb582ac74db12571ba2fccf28601b, 0xfa8d52f970ba51420be43501370b16, 0x001996 ] } , + BigNum { limbs: [ 0x4dee80538980eca8e45d0230d79fe6, 0xacc5d6a40c8de1f7cc23d4a0a0fa3d, 0x005a56 ] } , + BigNum { limbs: [ 0x8a65bc0f8e4d1ecdd815ee4d942bee, 0xa05ea5c70ef50d10f88231dde2f6e1, 0x003336 ] } , + BigNum { limbs: [ 0x333e46f0700ee03227ea10b26bd413, 0x06f483d66e532628df85d7c3f50e72, 0x0040b7 ] } , + BigNum { limbs: [ 0x56ab6223eff0406e175e44c490cbc0, 0x7a32e902f9e85187145dfaed820595, 0x0059dc ] } , + BigNum { limbs: [ 0x66f8a0dc0e6bbe91e8a1ba3b6f3441, 0x2d20409a835fe1b2c3aa0eb455ffbe, 0x001a11 ] } , + BigNum { limbs: [ 0x4492b6f43bb51eca7dadda50f555e4, 0x9040823a88e7b6a42eb341ff879cb0, 0x006025 ] } , + BigNum { limbs: [ 0x79114c0bc2a6e035825224af0aaa1d, 0x1712a762f4607c95a954c7a25068a3, 0x0013c8 ] } , + BigNum { limbs: [ 0xc85f7efbd6188bb08956f964517062, 0x2b7d27b4c23e09768a7087f0cc2629, 0x001356 ] } , + BigNum { limbs: [ 0xf54484042843734f76a9059bae8f9f, 0x7bd601e8bb0a29c34d9781b10bdf29, 0x006097 ] } , + BigNum { limbs: [ 0xae4a21d39a23bee9dc91a650a90e49, 0x59d76ca02af7f2f3e98766f8c9185c, 0x0016c3 ] } , + BigNum { limbs: [ 0x0f59e12c64384016236e58af56f1b8, 0x4d7bbcfd52504045ee80a2a90eecf7, 0x005d2a ] } , + BigNum { limbs: [ 0xc708b8b84ee699a565af7a806913be, 0x169d8ee087328d16baf2bdd5e94641, 0x002db7 ] } , + BigNum { limbs: [ 0xf69b4a47af75655a9a50847f96ec43, 0x90b59abcf615a6231d154bcbeebf11, 0x004636 ] } , + BigNum { limbs: [ 0xfb135ff413a356d5f57c79b59b2332, 0x73726e9409979c7943d6a33bc39345, 0x0061f2 ] } , + BigNum { limbs: [ 0xc290a30beab8a82a0a83854a64dccf, 0x33e0bb0973b096c09431666614720d, 0x0011fb ] } , + BigNum { limbs: [ 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0x000631 ] } , + BigNum { limbs: [ 0xe7613894a9b5c86333defbe1eafec8, 0xecb3aff7822dc90fd38cadbdeab57e, 0x0051a0 ] } , + BigNum { limbs: [ 0xd642ca6b54a6369ccc21031e150139, 0xba9f79a5fb1a6a2a047b5be3ed4fd4, 0x00224c ] } , + BigNum { limbs: [ 0xbd99d6f7d72c9825c9330c166c32b6, 0x656984a80d04d77f485529026c165c, 0x0008f3 ] } , + BigNum { limbs: [ 0x000a2c08272f66da36ccf2e993cd4b, 0x41e9a4f570435bba8fb2e09f6beef7, 0x006afa ] } , + BigNum { limbs: [ 0x00ac7d231f0dcc77f716ad84ec3f4e, 0x66b749227bc79c2b526b533ce6736d, 0x001236 ] } , + BigNum { limbs: [ 0xbcf785dcdf4e328808e9517b13c0b3, 0x409be07b0180970e859cb664f191e6, 0x0061b7 ] } , + BigNum { limbs: [ 0x44df453062977b35a87e2790b1aa63, 0x70a52543032792c60337c3de194e28, 0x004ca4 ] } , + BigNum { limbs: [ 0x78c4bdcf9bc483ca5781d76f4e559e, 0x36ae045a7a20a073d4d045c3beb72b, 0x002749 ] } , + BigNum { limbs: [ 0x009c43f83b178b774638c79cf5dc1f, 0x3f6e8235ecae3905159358e0d38ebb, 0x0010c4 ] } , + BigNum { limbs: [ 0xbd07bf07c3447388b9c737630a23e2, 0x67e4a7679099fa34c274b0c1047698, 0x006329 ] } , + BigNum { limbs: [ 0x2a876351c2a2215759eb2f01eda387, 0xcb3d2bb78a0443275a5062c289ae5a, 0x0051ae ] } , + BigNum { limbs: [ 0x931c9fae3bb9dda8a614cffe125c7a, 0xdc15fde5f343f0127db7a6df4e56f9, 0x00223e ] } , + BigNum { limbs: [ 0xa8a3a7f2912453021ed75861bf15b2, 0xb5e368fea545ec1883b4b3c9abedbf, 0x0052cb ] } , + BigNum { limbs: [ 0x15005b0d6d37abfde128a69e40ea4f, 0xf16fc09ed8024721545355d82c1794, 0x002121 ] } , + BigNum { limbs: [ 0x99bf3cadf47500466419f0acda4132, 0x0fab03bda4f931faf27e04e17cf31b, 0x004831 ] } , + BigNum { limbs: [ 0x23e4c65209e6feb99be60e5325becf, 0x97a825dfd84f013ee58a04c05b1238, 0x002bbc ] } , + BigNum { limbs: [ 0x2af33beac6f68086ebdd53f2e05b8c, 0x18dfadf4968e6c38c06e52f5b19228, 0x003f52 ] } , + BigNum { limbs: [ 0x92b0c71537657e791422ab0d1fa475, 0x8e737ba8e6b9c7011799b6ac26732b, 0x00349b ] } , + BigNum { limbs: [ 0x532ae018561e10623432bc97e66621, 0x7d758178f8009d3d7a7ff033103d20, 0x003c40 ] } , + BigNum { limbs: [ 0x6a7922e7a83dee9dcbcd42681999e0, 0x29dda824854795fc5d88196ec7c833, 0x0037ad ] } , + BigNum { limbs: [ 0x5fdb8562ecaee38095506aaf2ed4e8, 0x40fa9e92971b3b44484802d2908ee1, 0x005327 ] } , + BigNum { limbs: [ 0x5dc87d9d11ad1b7f6aaf9450d12b19, 0x66588b0ae62cf7f58fc006cf477672, 0x0020c6 ] } , + BigNum { limbs: [ 0xc00587548157407cc2c1b551422388, 0x2433aacc79664fd54b05535ed75859, 0x0006c4 ] } , + BigNum { limbs: [ 0xfd9e7bab7d04be833d3e49aebddc79, 0x831f7ed103e1e3648d02b64300acf9, 0x006d29 ] } , + BigNum { limbs: [ 0xabf9af77e7e83655b9012175a8cc3b, 0xe87fd6b51c9c4226a334d73bebcb2b, 0x000b72 ] } , + BigNum { limbs: [ 0x11aa53881673c8aa46fedd8a5733c6, 0xbed352e860abf11334d33265ec3a28, 0x00687a ] } , + BigNum { limbs: [ 0x1fbcdb53444abbab0181bccc152b72, 0xedc455355287c6bf314728f7672064, 0x0068e3 ] } , + BigNum { limbs: [ 0x9de727acba114354fe7e4233ead48f, 0xb98ed4682ac06c7aa6c0e0aa70e4ef, 0x000b09 ] } , + BigNum { limbs: [ 0x5603226dcddca5147d1e72d2b5d06e, 0xc12baea83c091f5a39e548082c8807, 0x0064a3 ] } , + BigNum { limbs: [ 0x67a0e092307f59eb82e18c2d4a2f93, 0xe6277af5413f13df9e22c199ab7d4c, 0x000f49 ] } , + BigNum { limbs: [ 0x2ed8500386c76648c8e50cb39616ec, 0xef6739da92bc569aa8da7c3ebc2a85, 0x005a31 ] } , + BigNum { limbs: [ 0x8ecbb2fc779498b7371af24c69e915, 0xb7ebefc2ea8bdc9f2f2d8d631bdace, 0x0019bb ] } , + BigNum { limbs: [ 0xfe3a2e7b95c6469630cf3b08b533d4, 0x2a8f2e2a85919ad23e75d282b2d3b6, 0x006802 ] } , + BigNum { limbs: [ 0xbf69d4846895b869cf30c3f74acc2d, 0x7cc3fb72f7b698679992371f25319c, 0x000beb ] } , + BigNum { limbs: [ 0xcae90dfb3c0064c6123fa45fb28c2e, 0xd4cb52a34e3d7519670d09634336d4, 0x0023ce ] } , + BigNum { limbs: [ 0xf2baf504c25b9a39edc05aa04d73d3, 0xd287d6fa2f0abe2070fb003e94ce7e, 0x00501e ] } , + BigNum { limbs: [ 0x5ca0f385906331980144927341bf91, 0xe145a4a45855dd7ae283294477734f, 0x006964 ] } , + BigNum { limbs: [ 0x61030f7a6df8cd67febb6c8cbe4070, 0xc60d84f924f255bef584e05d609204, 0x000a88 ] } , + BigNum { limbs: [ 0xa47df0bc824e8ea22755dfe7bb9750, 0xfb32f90ffda0d607ced8ce69bc11a9, 0x0046cf ] } , + BigNum { limbs: [ 0x192612437c0d705dd8aa1f184468b1, 0xac20308d7fa75d32092f3b381bf3aa, 0x002d1d ] } , + BigNum { limbs: [ 0xec958cdc54a63cc7a96c1c03278845, 0x14461e4a1131fa3b366dda348f3893, 0x000791 ] } , + BigNum { limbs: [ 0xd10e7623a9b5c2385693e2fcd877bc, 0x930d0b536c1638fea19a2f6d48ccbf, 0x006c5c ] } , + BigNum { limbs: [ 0xb5abf8c3e544b7cec40dd9d4019abd, 0xe3e21a204627ca43fa39cf3715caad, 0x002b0f ] } , + BigNum { limbs: [ 0x07f80a3c191747313bf2252bfe6544, 0xc3710f7d372068f5ddce3a6ac23aa6, 0x0048dd ] } , + BigNum { limbs: [ 0x40722db5a2fdf30997af02bca933c9, 0x56c300383c82d6ab35ba4d5db3f1f7, 0x00688a ] } , + BigNum { limbs: [ 0x7d31d54a5b5e0bf66850fc4356cc38, 0x5090296540c55c8ea24dbc4424135c, 0x000b63 ] } , + BigNum { limbs: [ 0x564fcbc247d6d6ac9cc344a8cfff5a, 0xeb803c23772e9c6afb16ff55ea304f, 0x004b12 ] } , + BigNum { limbs: [ 0x6754373db6852853633cba573000a7, 0xbbd2ed7a061996cedcf10a4bedd504, 0x0028da ] } , + BigNum { limbs: [ 0xcdd1364d8cfacb93f88bc67b7b3721, 0x665f2b83a6deb7fdceb13345570e65, 0x0022a4 ] } , + BigNum { limbs: [ 0xefd2ccb27161336c0774388484c8e0, 0x40f3fe19d6697b3c0956d65c80f6ed, 0x005149 ] } , + BigNum { limbs: [ 0x0e9d35b6ed76adf7713ef284c069ff, 0xcf43948b469bcd431486dfe074f000, 0x002d69 ] } , + BigNum { limbs: [ 0xaf06cd4910e551088ec10c7b3f9602, 0xd80f951236ac65f6c38129c1631553, 0x004683 ] } , + BigNum { limbs: [ 0x67d80e0380c9dd297e9abd55f848ed, 0x70bfd3999432941131a1638c47aaf6, 0x002ffb ] } , + BigNum { limbs: [ 0x55cbf4fc7d9221d6816541aa07b714, 0x36935603e9159f28a666a615905a5d, 0x0043f2 ] } , + BigNum { limbs: [ 0x8032cefe4d368500c64cd40384f439, 0xe7910a9571c7e580abd9bdea663091, 0x001d7f ] } , + BigNum { limbs: [ 0x3d713401b12579ff39b32afc7b0bc8, 0xbfc21f080b804db92c2e4bb771d4c2, 0x00566d ] } , + BigNum { limbs: [ 0x121d901c68c49123c031d18d8c2649, 0xd8bc987365098ce1593a825ad83ef5, 0x006fc8 ] } , + BigNum { limbs: [ 0xab8672e395976ddc3fce2d7273d9b8, 0xce96912a183ea6587ecd8746ffc65e, 0x000424 ] } , + BigNum { limbs: [ 0xe9d91c5935fd49f4b6cc71968e9630, 0xab942a7f28f2aeb5acc56ef4dd0686, 0x003414 ] } , + BigNum { limbs: [ 0xd3cae6a6c85eb50b49338d697169d1, 0xfbbeff1e545584842b429aacfafecc, 0x003fd8 ] } , + BigNum { limbs: [ 0x40d3a6dad17cfb58253c958a750acb, 0xd11e6709e3150b19cc92362eeaac79, 0x00503d ] } , + BigNum { limbs: [ 0x7cd05c252cdf03a7dac369758af536, 0xd634c2939a3328200b75d372ed58da, 0x0023af ] } , + BigNum { limbs: [ 0x25871b96c0742e75161473966a53f7, 0x032dedb0b892480e75bd26e4e2ccb3, 0x000e42 ] } , + BigNum { limbs: [ 0x981ce7693de7d08ae9eb8b6995ac0a, 0xa4253becc4b5eb2b624ae2bcf538a0, 0x0065ab ] } , + BigNum { limbs: [ 0x309054dce40022360156e8b623013e, 0xbaeda02a6ff092a7ec5a61b18bafed, 0x003381 ] } , + BigNum { limbs: [ 0x8d13ae231a5bdcc9fea91649dcfec3, 0xec6589730d57a091ebada7f04c5566, 0x00406b ] } , + BigNum { limbs: [ 0xa9d4bab0b502dd86edbea447c942d5, 0x0a33a25ac40a26f8483151cc7c8088, 0x0060a2 ] } , + BigNum { limbs: [ 0x13cf484f4959217912415ab836bd2c, 0x9d1f8742b93e0c418fd6b7d55b84cb, 0x00134b ] } , + BigNum { limbs: [ 0x269bddbc2b9df4ce9a534745bfd613, 0xeb58904936dbdf2b8960761678b86c, 0x001fa4 ] } , + BigNum { limbs: [ 0x97082543d2be0a3165acb7ba4029ee, 0xbbfa9954466c540e4ea7938b5f4ce7, 0x005448 ] } , + BigNum { limbs: [ 0xc4c8739d4475765bf0e8fec7a9af22, 0xd3a3929c4bb36342213767f33397b9, 0x002c4c ] } , + BigNum { limbs: [ 0xf8db8f62b9e688a40f1700385650df, 0xd3af97013194cff7b6d0a1aea46d99, 0x0047a0 ] } , + BigNum { limbs: [ 0x8c7a99da8938647073084001146f23, 0x948861351141f0f5093f56be4518ad, 0x000b04 ] } , + BigNum { limbs: [ 0x3129692575239a8f8cf7befeeb90de, 0x12cac8686c064244cec8b2e392eca6, 0x0068e9 ] } , + BigNum { limbs: [ 0xab88365b6035748de74346b283c145, 0xa824ff74aa74280674d32da105c40d, 0x004f97 ] } , + BigNum { limbs: [ 0x121bcca49e268a7218bcb84d7c3ebc, 0xff2e2a28d2d40b336334dc00d24146, 0x002455 ] } , + BigNum { limbs: [ 0x76d28a9be629fb69e5dad127abcc9f, 0x7efc6393c8cfe857780d4fc57e7f03, 0x003ad2 ] } , + BigNum { limbs: [ 0x46d17864183203961a252dd8543362, 0x2856c609b4784ae25ffab9dc598650, 0x00391b ] } +]; diff --git a/noir-projects/noir-protocol-circuits/crates/blob/src/main.nr b/noir-projects/noir-protocol-circuits/crates/blob/src/main.nr new file mode 100644 index 00000000000..fde8ce1beda --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/src/main.nr @@ -0,0 +1,616 @@ +// ONLY IMPORT ONE OF THESE CONFIGS! The big `config` takes 11 mins to compile. + +// SMALL CONFIG ********************************************************************************* + +// mod smaller_config; + +// use crate::smaller_config::{ +// BigNum, BLS12_381_Fr_Params, F, FIELDS_PER_BLOB, LOG_FIELDS_PER_BLOB, NOIR_FIELDS_PER_BLOB, +// FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB, D, D_INV, ROOTS, NEGATIVE_ROOTS +// }; + +//********************************************************************************* + +// BIG CONFIG ********************************************************************* + +mod config; +mod negative_roots; + +use crate::{ + config::{ + BigNum, BLS12_381_Fr_Params, F, FIELDS_PER_BLOB, LOG_FIELDS_PER_BLOB, NOIR_FIELDS_PER_BLOB, + FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB, D, D_INV, ROOTS +}, + negative_roots::NEGATIVE_ROOTS +}; + +//********************************************************************************* + +use std::hash::poseidon2; + +global LIMB_MAX = 2.pow_32(120); + +unconstrained fn __batch_invert_impl(mut x: [F; N]) -> [F; N] { + let mut accumulator: F = BigNum::one(); + + let mut temporaries: [F] = &[]; + for i in 0..x.len() { + temporaries = temporaries.push_back(accumulator); + if (x[i].__is_zero() == false) { + accumulator = accumulator.__mul(x[i]); + } + } + + accumulator = accumulator.__invmod(); + let mut T0: F = BigNum::new(); + for i in 0..x.len() { + let idx = x.len() - 1 - i; + if (x[idx].__is_zero() == false) { + T0 = accumulator.__mul(temporaries[idx]); + accumulator = accumulator.__mul(x[idx]); + x[idx] = T0; + } + } + x +} + +// Not used because it resulted in "stack too deep", so it's inlined instead. +unconstrained fn __compute_fracs(z: F, ys: [F; FIELDS_PER_BLOB]) -> [F; FIELDS_PER_BLOB] { + let mut denoms: [F; FIELDS_PER_BLOB] = [BigNum::new(); FIELDS_PER_BLOB]; + for i in 0..FIELDS_PER_BLOB { + denoms[i] = z.__add(NEGATIVE_ROOTS[i]); // (z - ω^i) + } + let inv_denoms = __batch_invert_impl(denoms); // 1 / (z - ω^i), for all i + + let mut fracs: [F; FIELDS_PER_BLOB] = [BigNum::new(); FIELDS_PER_BLOB]; + for i in 0..FIELDS_PER_BLOB { + let inv_denom = inv_denoms[i]; // 1 / (z - ω^i) + + fracs[i] = ys[i].__mul(inv_denom); // y_i / (z - ω^i) + } + fracs +} + +unconstrained fn __field_to_bytes(x: Field) -> [u8; 32] { + let x_bytes_slice = x.to_be_bytes(32); + let mut x_bytes = [0; 32]; + for i in 0..32 { + x_bytes[i] = x_bytes_slice[i]; + } + x_bytes +} + +unconstrained fn __field_to_bignum(x: Field) -> F { + let x_bytes = __field_to_bytes(x); + + BigNum::from_byte_be(x_bytes) + // TODO: use this once the name has been updated in the bignum lib: + // BigNum::from_be_bytes(x_bytes) +} + +unconstrained fn __field_to_bignum_limbs(x: Field) -> [Field; 3] { + __field_to_bignum(x).limbs +} + +// Only works for bignums with modulus larger than the BN Fr size (which is true +// for the bls12-381 Fr field). +fn field_to_bignum(x: Field) -> F { + let __x_limbs = __field_to_bignum_limbs(x); + + let mut check = __x_limbs[3 - 1]; + for i in 1..3 { + check *= LIMB_MAX; + check += __x_limbs[3 - i - 1]; + } + assert(check == x); + + BigNum { limbs: __x_limbs } +} + +// DANGER: this assumes the input bignum is <= the Noir field size. +// Only use this if you _know_ the data being passed in is small enough. +// +// Or actually, maybe it's not unsafe, if Field catches overflows? +fn unsafe_bignum_to_field(x: F) -> Field { + let mut result: Field = 0; + result += x.limbs[3 - 1]; + for i in 1..3 { + result *= LIMB_MAX; + result += x.limbs[3 - i - 1]; + } + result +} + +fn bignum_to_bytes(x: F) -> [u8] { + let limb_0_bytes: [u8] = x.limbs[0].to_be_bytes(15); + let limb_1_bytes: [u8] = x.limbs[1].to_be_bytes(15); + let limb_2_bytes: [u8] = x.limbs[2].to_be_bytes(2); + let out = limb_2_bytes.append(limb_1_bytes).append(limb_0_bytes); + std::static_assert(out.len() == 32, "bad byte decomposition of bignum"); + out +} + +// fn kzg_commitment_to_bytes(c: [Field; 2]) -> [u8] { +// let limb_0_bytes: [u8] = x.limbs[0].to_be_bytes(32); +// let limb_1_bytes: [u8] = x.limbs[1].to_be_bytes(16); + +// let out = limb_2_bytes.append(limb_1_bytes).append(limb_0_bytes); +// std::static_assert(out.len() == 32, "bad byte decomposition of bignum"); +// out +// } + +// DANGER: this assumes the input bignum is <= the Noir field size. +// Only use this if you _know_ the data being passed in is small enough. +// +// This is inefficient, in the sense that we discard ~1 bit of blob space per +// 255-bit blob field, when converting it to a 245-bit noir field. Over the whole blob, +// we end up discarding 1 bit * 4096 fields_per_blob = 512 bytes = 16 words of data. +// BUT, it is much simpler to do this than to reconstitute 4096 disparate bits across +// the whole blob into 16 words. Perhaps the more complex approach should only be +// taken once aztec blobs are sufficiently full? +fn unsafe_blob_to_fields(blob: [F; FIELDS_PER_BLOB]) -> [Field; FIELDS_PER_BLOB] { + let mut blob_as_fields: [Field; FIELDS_PER_BLOB] = [0; FIELDS_PER_BLOB]; + for i in 0..FIELDS_PER_BLOB { + blob_as_fields[i] = unsafe_bignum_to_field(blob[i]); + } + blob_as_fields +} + +unconstrained fn __unsafe_extract_top_bit(x: F) -> (Field, Field) { + let top_limb: Field = x.limbs[2]; + // The top_limb is at most 2 bytes (16 bits). + // 0x8000 = 2^15 = 32768 + let top_bit: Field = (top_limb as u16 / 0x8000) as Field; + let top_limb_with_top_bit_removed = top_limb - top_bit * 0x8000; + (top_bit, top_limb_with_top_bit_removed) +} + +// DANGER: it's named as "unsafe" because the caller MUST already have checked that +// each blob Field is formatted as (u1, Field). I.e. the "rhs" 254-bits should already +// fit within a Field. If the "rhs" 254 bits is larger than the field modulus, +// there will be an uncaught overflow of the 254-bits in the Field, resulting in +// an unintended tiny value. +// +// For efficiency, the top_bit is kept as a Field throughout. +fn unsafe_extract_top_bit(x: F) -> (Field, F) { + let (top_bit, top_limb_with_top_bit_removed) = __unsafe_extract_top_bit(x); + assert_eq(top_bit * 0x8000 + top_limb_with_top_bit_removed, x.limbs[2]); + + (top_bit, BigNum { limbs: [x.limbs[0], x.limbs[1], top_limb_with_top_bit_removed] }) +} + +fn blob_to_fields__tightly_packed(blob: [F; FIELDS_PER_BLOB]) -> [Field; NOIR_FIELDS_PER_BLOB] { + let mut blob_as_fields: [Field; NOIR_FIELDS_PER_BLOB] = [0; NOIR_FIELDS_PER_BLOB]; + let mut top_bits: [Field; FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB] = [0; FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB]; + + // We start with [F; 4096]. + // The first 4064 of these bls-fields have a 255th bit (counting from 1) which can contribute towards + // new 254-bit noir fields. That is, we extract 4064 top-bits from the first 4064 of the 4096 bls-fields, + // and reconstitute them into 4064 / 254 = 16 extra noir fields. + // So we end up with 4096 + 16 = 4112 noir fields. + + // Here we compute top_bits[0:4064] and blob_as_fields[0:4064]. + for i in 0..FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB { + let (top_bit, field_with_top_bit_removed): (Field, F) = unsafe_extract_top_bit(blob[i]); + top_bits[i] = top_bit; + blob_as_fields[i] = unsafe_bignum_to_field(field_with_top_bit_removed); + } + // Here we compute blob_as_fields[4064:4096]. + for i in FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB..FIELDS_PER_BLOB { + blob_as_fields[i] = unsafe_bignum_to_field(blob[i]); + } + // Here we compute blob_as_fields[4096:4112] from top_bits[0:4064] + for i in FIELDS_PER_BLOB..NOIR_FIELDS_PER_BLOB { + // the top_bits are assumed to be big-endian bit arrays: + let mut reconstituted_field = top_bits[0]; + for j in 1..254 { + let k = (i - FIELDS_PER_BLOB) * 254 + j; + reconstituted_field *= 2; + reconstituted_field += top_bits[k]; + // std::as_witness(reconstituted_field); // this was costing 4048 gates + } + blob_as_fields[i] = reconstituted_field; + } + blob_as_fields +} + +// TODO: We'll want to hash this data +// in an arrangement which makes sense to the aztec protocol. THink about this more. +fn hash_blob(blob: [F; FIELDS_PER_BLOB]) -> Field { + // let mut blob_as_fields = unsafe_blob_to_fields(blob); + let mut blob_as_fields = blob_to_fields__tightly_packed(blob); + let hash = poseidon2::Poseidon2::hash(blob_as_fields, blob_as_fields.len()); + hash +} + +fn hash_kzg_commitment(kzg_commitment: [Field; 2]) -> Field { + let hash = poseidon2::Poseidon2::hash(kzg_commitment, kzg_commitment.len()); + hash +} + +fn compute_challenge(blob: [F; FIELDS_PER_BLOB], kzg_commitment: [Field; 2]) -> Field { + let kzg_commitment_hash = hash_kzg_commitment(kzg_commitment); + let blob_hash = hash_blob(blob); + // let blob_hash = 1; + let challenge = poseidon2::Poseidon2::hash([blob_hash, kzg_commitment_hash], 2); + challenge +} + +// ~500k constraints. 30 MINUTES TO COMPILE (due to all the brillig)! +// +// Note: the kzg_commitment is technically a BLS12-381 point in (Fq, Fq), but +// we haven't implemented Fq; only Fr, and we don't actually need to operate on it; +// we just need the bits of data. So we've simply encoded it as fitting inside a +// [Field; 2], since two 254-bit fields more-than covers 381+1=382 bits. +fn main(blob: [F; FIELDS_PER_BLOB], kzg_commitment: [Field; 2]) -> pub (Field, F, [Field; 2]) { + let challenge_z: Field = compute_challenge(blob, kzg_commitment); + let challenge_z_as_bignum: F = field_to_bignum(challenge_z); + + let y: F = barycentric_evaluate_blob_at_z(challenge_z_as_bignum, blob); + // let y = challenge_z_as_bignum; // dummy constraint for when we want to skip the barycentric stuff in testing. + + // let challenge_z_as_bytes: [u8] = challenge_z.to_be_bytes(32); + // let y_as_bytes: [u8] = bignum_to_bytes(y); + // let kzg_commitment_as_bytes: [u8] = () + + // TODO: this return data needs to be TIGHTLY PACKED into bytes. + // TODO: then those bytes need to be sha256-hashed, to produce a single value that can be sent to ethereum for cheap snark verification. On ethereum, the bytes will be sent along with the sha256-hash of the bytes. The bytes will be used in the point evaluation precompile. The sha256-hash will form a part of the public inputs of the zk-snark proof. + (challenge_z, y, kzg_commitment) +} + +/** + * ___d-1 + * z^d - 1 \ ω^i + * p(z) = --------- . / y_i . --------- + * d /____ z - ω^i + * i=0 + * + * p(z) = factor . sum( y_i . num / denom ) + * + * + * where d = 4096 + * + * Precompute: + * - The d roots of unity ω^i (plus maybe their negatives for z - ω^i computations). + * - (1 / d) + * + * @param z + * @param ys - the many y_i's of the blob. + * + * @return y = p(z) + */ +fn barycentric_evaluate_blob_at_z(z: F, ys: [F; FIELDS_PER_BLOB]) -> F { + // z ^ D: + + let mut t1 = z.__mul(z); + + BigNum::evaluate_quadratic_expression([[z]], [[false]], [[z]], [[false]], [t1], [true]); + + let mut t2: F = BigNum::new(); + for _i in 0..LOG_FIELDS_PER_BLOB - 1 { + t2 = t1.__mul(t1); + + // GRATUITOUS USAGE OF as_witness, LIKE THROWING DARTS AT A DARTBOARD AND HOPING THIS HELPS + std::as_witness(t2.limbs[0]); + std::as_witness(t2.limbs[1]); + std::as_witness(t2.limbs[2]); + + BigNum::evaluate_quadratic_expression([[t1]], [[false]], [[t1]], [[false]], [t2], [true]); + + t1 = t2; + std::as_witness(t1.limbs[0]); + std::as_witness(t1.limbs[1]); + std::as_witness(t1.limbs[2]); + } + + let z_pow_d = t1; + + // factor: + + let one: F = BigNum::one(); + + t1 = z_pow_d.__sub(one); + std::as_witness(t1.limbs[0]); + std::as_witness(t1.limbs[1]); + std::as_witness(t1.limbs[2]); + + let factor = t1.__mul(D_INV); + + // (z_pow_d - one) * (D_INV) - factor = 0 + // z_pow_d * D_INV - D_INV - factor = 0 + BigNum::evaluate_quadratic_expression( + [[z_pow_d]], + [[false]], + [[D_INV]], + [[false]], + [factor, D_INV], + [true, true] + ); + + // This version doesn't work: + // BigNum::evaluate_quadratic_expression( + // [[z_pow_d, one]], + // [[false, true]], + // [[D_INV]], + // [[false]], + // [factor], + // [true] + // ); + + // sum: + + let mut sum: F = BigNum::new(); + + // Making a call to this function causes a "stack too deep" error, so I've put the body of that function here, instead: + // let fracs = __compute_fracs(z, ys); // { y_i / (z - ω^i) } + + /** + * + * Note: it's more efficient (saving 30k constraints) to compute: + * ___d-1 + * \ / y_i \ + * / | --------- | . ω^i + * /____ \ z - ω^i / + * i=0 + * ^^^^^^^^^ + * frac + * + * ... than to compute: + * + * ___d-1 + * \ / ω^i \ + * / y_i . | --------- | + * /____ \ z - ω^i / + * i=0 + * + * perhaps because all the ω^i terms are constant witnesses? + * + */ + + //***************************************************************** + // This section is only needed because `__compute_fracs` isn't working (stack too deep error). + + let mut fracs: [F; FIELDS_PER_BLOB] = [BigNum::new(); FIELDS_PER_BLOB]; // y_i / (z - ω^i), for all i + + let mut denoms = [BigNum::new(); FIELDS_PER_BLOB]; + for i in 0..FIELDS_PER_BLOB { + denoms[i] = z.__add(NEGATIVE_ROOTS[i]); // (z - ω^i) + } + + // If you're seeing a `bug` warning for this line, I think it's fine. + // Ideally, we'd be using `__compute_fracs`, anyway, but we're getting a "stack too deep" error. + let inv_denoms = __batch_invert_impl(denoms); // 1 / (z - ω^i), for all i + + for i in 0..FIELDS_PER_BLOB { + let num = ys[i]; + let inv_denom = inv_denoms[i]; // 1 / (z - ω^i) + + let frac = num.__mul(inv_denom); // y_i * (1 / (z - ω^i)) + + fracs[i] = frac; // y_i / (z - ω^i) + std::as_witness(fracs[i].limbs[0]); + std::as_witness(fracs[i].limbs[1]); + std::as_witness(fracs[i].limbs[2]); + + //End of section that is only needed because `__compute_fracs` isn't working + //***************************************************************** + + // frac <-- ys[i] / (z + neg_roots[i]) + // frac * (z + neg_roots[i]) - ys[i] = 0 + BigNum::evaluate_quadratic_expression( + [[fracs[i]]], + [[false]], + [[z, NEGATIVE_ROOTS[i]]], + [[false, false]], + [ys[i]], + [true] + ); + } + + // OK so...we can add multiple product terms into a sum...but I am not sure how many! + // we are computing 254 * 254 bit products and we need to ensure each product limb doesn't overflow + // each limb is 120 bits => 120 * 120 = 240 bits. + // however when computing a mul we add up to 5 product terms into a single field element => 243 bits (ish) + // when we do a modular reduction we validate that a field element >> 120 bits is less than 2^{126} which implies we have 246 bits to play with + // which implies...we can accomodate up to EIGHT additions of product terms before we risk overflowing + // (this is really messy! I never considered the case of giant linear sequences of products) + let mut sum: F = BigNum::new(); + /** + * Seeking: + * ___d-1 + * \ ω^i + * sum = / y_i . --------- + * /____ z - ω^i + * i=0 + */ + let NUM_PARTIAL_SUMS = FIELDS_PER_BLOB / 8; + for i in 0..NUM_PARTIAL_SUMS { + let mut partial_sum: F = BigNum::new(); + let mut lhs: [F; 8] = [BigNum::new(); 8]; + let mut rhs = lhs; + + /** + * Seeking: + * ___i*8 + 7 + * \ ω^k + * partial_sum = / y_k . --------- + * /____ z - ω^k + * k=i*8 + 0 + */ + + for j in 0..8 { + let k = i * 8 + j; + lhs[j] = ROOTS[k]; // ω^k + rhs[j] = fracs[k]; // y_k / (z - ω^k) + std::as_witness(lhs[j].limbs[0]); + std::as_witness(lhs[j].limbs[1]); + std::as_witness(lhs[j].limbs[2]); + std::as_witness(rhs[j].limbs[0]); + std::as_witness(rhs[j].limbs[1]); + std::as_witness(rhs[j].limbs[2]); + + // y_k * ( ω^k / (z - ω^k) ) + let summand = ROOTS[k].__mul(fracs[k]); + + // partial_sum + ( y_k * ( ω^k / (z - ω^k) ) -> partial_sum + partial_sum = partial_sum.__add(summand); + std::as_witness(partial_sum.limbs[0]); + std::as_witness(partial_sum.limbs[1]); + std::as_witness(partial_sum.limbs[2]); + } + + /** + * Seeking: + * ___i*8 - 1 ___i*8 + 7 + * \ ω^i \ / y_k \ + * sum_out = / y_i . --------- + / ω^k . | --------- | + * /____ z - ω^i /____ \ z - ω^k / + * 0 k = i*8 + * ^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + * sum partial_sum + * + * ... that is: + * + * ___i*8 - 1 ___ 7 + * \ ω^i \ + * sum_out = / y_i . --------- + / lhs[j] . rhs[j] + * /____ z - ω^i /____ + * 0 j = 0 + * ^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^ + * sum partial_sum + */ + + let mut sum_out = sum.__add(partial_sum); + + std::as_witness(sum_out.limbs[0]); + std::as_witness(sum_out.limbs[1]); + std::as_witness(sum_out.limbs[2]); + + // sum_out <- sum + (lhs[0] * rhs[0] + ... + lhs[7] * rhs[7]) + // => (lhs[0] * rhs[0] + ... + lhs[7] * rhs[7]) + sum - sum_out == 0 + BigNum::evaluate_quadratic_expression( + [[lhs[0]], [lhs[1]], [lhs[2]], [lhs[3]], [lhs[4]], [lhs[5]], [lhs[6]], [lhs[7]]], + [[false], [false], [false], [false], [false], [false], [false], [false]], + [[rhs[0]], [rhs[1]], [rhs[2]], [rhs[3]], [rhs[4]], [rhs[5]], [rhs[6]], [rhs[7]]], + [[false], [false], [false], [false], [false], [false], [false], [false]], + [sum, sum_out], + [false, true] + ); + + sum = sum_out; + std::as_witness(sum.limbs[0]); + std::as_witness(sum.limbs[1]); + std::as_witness(sum.limbs[2]); + } + + // y: + + let y = factor.__mul(sum); + + // y <- factor * sum + // => factor * sum - y == 0 + BigNum::evaluate_quadratic_expression([[factor]], [[false]], [[sum]], [[false]], [y], [true]); + + println(y); + + y +} + +// Need to wait for https://github.com/noir-lang/noir-bignum/pull/9 for this to work. +// nargo test --show-output test_main +// #[test] +// fn test_main() { +// let mut blob: [F; FIELDS_PER_BLOB] = [BigNum::new(); FIELDS_PER_BLOB]; + +// blob[0] = BigNum { limbs: [0x1234, 0, 0] }; +// blob[1] = BigNum { limbs: [0xabcd, 0, 0] }; +// blob[2] = BigNum { limbs: [0x69, 0, 0] }; + +// let kzg_commitment_in = [1, 2]; // this is made-up nonsense. + +// let (challenge_z, y, kzg_commitment) = main(blob, kzg_commitment_in); + +// println(challenge_z); +// println(y); +// println(kzg_commitment); +// } + +#[test] +fn test_print() { + let x = 4; + println(f"x: {x}"); +} + +// Need to wait for https://github.com/noir-lang/noir-bignum/pull/9 for this to work. +// #[test] +// fn test_bignum_conversions() { +// let x = 1000; +// let x_bignum = __field_to_bignum(x); +// println(f"x_bignum: {x_bignum}"); +// } + +// nargo test --show-output test_barycentric +#[test] +fn test_barycentric() { + let z: F = BigNum { limbs: [2, 0, 0] }; + + // many y's form a blob: + let mut ys: [F; FIELDS_PER_BLOB] = [BigNum::new(); FIELDS_PER_BLOB]; + + ys[0] = BigNum { limbs: [0x1234, 0, 0] }; + ys[1] = BigNum { limbs: [0xabcd, 0, 0] }; + ys[2] = BigNum { limbs: [0x69, 0, 0] }; + + // evaluate the blob at z = 2 to yield y: + let y = barycentric_evaluate_blob_at_z(z, ys); + + let mut expected_y: [Field; 3] = [0; 3]; + if (FIELDS_PER_BLOB == 4096) { + // Computed with the eth consensus specs py lib + expected_y = [ + 0x0c62e352a428e8e9842eadc1c106bd, + 0x902c5b4968d755b6f49c0231e15af8, + 0x00049a + ]; + // Also computed with cKzg, in the typescript tests: + // 0x049a902c5b4968d755b6f49c0231e15af80c62e352a428e8e9842eadc1c106bd + } + if (FIELDS_PER_BLOB == 8) { + // Computed with the eth consensus specs py lib (after hacking it to cope with blobs of size 8 instead of 4096): + expected_y = [ + 0xb04cdea4304000053abffffffb203a, + 0x0000000002e30785c8afa4496f8e38, + 0x000000 + ]; + } + assert(y.limbs == expected_y); +} + +// Helper function used to populate the hard-coded double_modulus value in the bls12381Fr.nr file in the bignum library. +unconstrained fn compute_double_modulus() -> [Field; 3] { + let two_p = [0x7b4805fffcb7fdfffffffe00000002, 0x4ea6533afa906673b0101343b00aa7, 0x00e7db]; + let NUM_LIMBS = 3; // must be >= 3 + let two_pow_120 = 2.pow_32(120); + let mut double_modulus: [Field; 3] = [0; 3]; + + double_modulus[0] = two_p[0] + two_pow_120; + for i in 1..NUM_LIMBS - 1 { + double_modulus[i] = two_p[i] + two_pow_120 - 1; + } + double_modulus[NUM_LIMBS - 1] = two_p[NUM_LIMBS - 1] - 1; + double_modulus +} + +// nargo test --show-output test_compute_double_modulus +#[test] +fn test_compute_double_modulus() { + println(compute_double_modulus()); +} + +// nargo test --show-output test_compute_d_inv +#[test] +fn test_compute_d_inv() { + let D_INV = D.__invmod(); + println(D_INV); +} + diff --git a/noir-projects/noir-protocol-circuits/crates/blob/src/negative_roots.nr b/noir-projects/noir-protocol-circuits/crates/blob/src/negative_roots.nr new file mode 100644 index 00000000000..a494173e618 --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/src/negative_roots.nr @@ -0,0 +1,4102 @@ +use dep::bigint::BigNum; + +// TODO: these negative roots are actually a rearrangement of the roots. If it is inefficient to load all 4096 of these bls fields (12k noir fields), we could perhaps modify the barycentric formula to do some clever indexing to lookup negative roots from the existing roots array. I'd be weary of auditability and bugs with such an approach, though. + +global NEGATIVE_ROOTS = [ + BigNum { limbs: [ 0xbda402fffe5bfeffffffff00000000, 0xa753299d7d483339d80809a1d80553, 0x0073ed ] } , + BigNum { limbs: [ 0x000000000000000000000000000001, 0x000000000000000000000000000000, 0x000000 ] } , + BigNum { limbs: [ 0xbaa40089fb5bfefffeffff00000001, 0xa753299d7d47a5e80b39939ed33467, 0x0073ed ] } , + BigNum { limbs: [ 0x030002760300000001000000000000, 0x0000000000008d51ccce760304d0ec, 0x000000 ] } , + BigNum { limbs: [ 0x32824078eaa4fe8dd702cb688bc087, 0x405d25a31660a733b23a98ca5b22a0, 0x003f96 ] } , + BigNum { limbs: [ 0x8b21c28713b7007228fd3397743f7a, 0x66f603fa66e78c0625cd70d77ce2b3, 0x003457 ] } , + BigNum { limbs: [ 0x4913cc6b4e0c269fc8017d5166afd3, 0xf524ccbc6d03787d7d083f1b189fc5, 0x0060b9 ] } , + BigNum { limbs: [ 0x74903694b04fd86037fe81ae99502e, 0xb22e5ce11044babc5affca86bf658e, 0x001333 ] } , + BigNum { limbs: [ 0xfef0f00df2ec88ac159e2688bd4333, 0xd8c1e977024e561dcd0fd4d314d93b, 0x00533b ] } , + BigNum { limbs: [ 0xbeb312f20b6f7653ea61d87742bcce, 0xce9140267af9dd1c0af834cec32c17, 0x0020b1 ] } , + BigNum { limbs: [ 0xc8bd90143c7a436a20b4c8fbee018e, 0x4de4b45f2d7bc4a72e43a8f20dbb34, 0x0024c1 ] } , + BigNum { limbs: [ 0xf4e672ebc1e1bb95df4b360411fe73, 0x596e753e4fcc6e92a9c460afca4a1e, 0x004f2c ] } , + BigNum { limbs: [ 0x43389604eeaf5a7f847ee47dc08d74, 0x15b4607e449bd66c91d61832fc60bd, 0x005511 ] } , + BigNum { limbs: [ 0x7a6b6cfb0faca4807b811a823f728d, 0x919ec91f38ac5ccd4631f16edba496, 0x001edc ] } , + BigNum { limbs: [ 0x010d53ccb225425e544cdf4c887948, 0xb475ab91194b687a73c92f188612fc, 0x003b25 ] } , + BigNum { limbs: [ 0xbc96af334c36bca1abb31fb37786b9, 0xf2dd7e0c63fccabf643eda8951f257, 0x0038c7 ] } , + BigNum { limbs: [ 0x4f1b07b3c56074c9f39f658c9620b3, 0x17191423f48d7e7d03f9e6ac83bc94, 0x00230d ] } , + BigNum { limbs: [ 0x6e88fb4c38fb8a360c60997369df4e, 0x903a157988bab4bcd40e22f55448bf, 0x0050e0 ] } , + BigNum { limbs: [ 0x0124a02ec3ede2c13af36ece5c935e, 0xe1cface780a62d34ccafd6d0ce6e2d, 0x000df6 ] } , + BigNum { limbs: [ 0xbc7f62d13a6e1c3ec50c9031a36ca3, 0xc5837cb5fca206050b5832d1099726, 0x0065f6 ] } , + BigNum { limbs: [ 0x7fdf9c77487ae79733f3a6ba60eaa6, 0xa2fb6162ffabd84f8612c8b6cc00bd, 0x00476f ] } , + BigNum { limbs: [ 0x3dc46688b5e11768cc0c58459f155b, 0x0457c83a7d9c5aea51f540eb0c0496, 0x002c7e ] } , + BigNum { limbs: [ 0xd460a1dbfcd267df2b12e5b0330148, 0xcd3a7fca77e38a490835c612d67951, 0x0020e9 ] } , + BigNum { limbs: [ 0xe943612401899720d4ed194fccfeb9, 0xda18a9d30564a8f0cfd2438f018c01, 0x005303 ] } , + BigNum { limbs: [ 0x8c291cbae2a361196986abd51edcb5, 0x83d4d0a0ae73ac3f6e2f24bd52fe70, 0x006f8c ] } , + BigNum { limbs: [ 0x317ae6451bb89de69679532ae1234c, 0x237e58fcced486fa69d8e4e48506e3, 0x000461 ] } , + BigNum { limbs: [ 0x6466930923ab7811a43132b1888c36, 0xf5e67a06fbd88f907adac98af5480e, 0x006f70 ] } , + BigNum { limbs: [ 0x593d6ff6dab086ee5bcecc4e7773cb, 0xb16caf96816fa3a95d2d4016e2bd45, 0x00047c ] } , + BigNum { limbs: [ 0x16d26dfeb2f7ee7d19743e43af5772, 0x4b9a3c48cf47ecaed406f9fc1065f1, 0x001cfa ] } , + BigNum { limbs: [ 0xa6d195014b641082e68bc0bc50a88f, 0x5bb8ed54ae00468b04010fa5c79f62, 0x0056f3 ] } , + BigNum { limbs: [ 0x7876bf0928e68fae34a81e1cfca2ec, 0xed8c432405d0ae25b21df6c52b2a6c, 0x005e73 ] } , + BigNum { limbs: [ 0x452d43f6d5756f51cb57e0e3035d15, 0xb9c6e6797777851425ea12dcacdae7, 0x001579 ] } , + BigNum { limbs: [ 0x2724b71d050aa97ebf2fcc0f5611ae, 0x440d3d981efb1e97f596a4c8da4826, 0x002e3e ] } , + BigNum { limbs: [ 0x967f4be2f951558140d032f0a9ee53, 0x6345ec055e4d14a1e27164d8fdbd2d, 0x0045af ] } , + BigNum { limbs: [ 0xcb9993a1e752756220b9443bf5371d, 0x77449923ce3ca1cac574d6e7aa0947, 0x004b02 ] } , + BigNum { limbs: [ 0xf20a6f5e1709899ddf46bac40ac8e4, 0x300e9079af0b916f129332ba2dfc0b, 0x0028eb ] } , + BigNum { limbs: [ 0xe1991fee0e510a7b9610b1db94977e, 0xf9e8afe7612dc1649310899759ba6e, 0x00205b ] } , + BigNum { limbs: [ 0xdc0ae311f00af48469ef4d246b6883, 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0x2ed8500386c76648c8e50cb39616ec, 0xef6739da92bc569aa8da7c3ebc2a85, 0x005a31 ] } , + BigNum { limbs: [ 0xbf69d4846895b869cf30c3f74acc2d, 0x7cc3fb72f7b698679992371f25319c, 0x000beb ] } , + BigNum { limbs: [ 0xfe3a2e7b95c6469630cf3b08b533d4, 0x2a8f2e2a85919ad23e75d282b2d3b6, 0x006802 ] } , + BigNum { limbs: [ 0xf2baf504c25b9a39edc05aa04d73d3, 0xd287d6fa2f0abe2070fb003e94ce7e, 0x00501e ] } , + BigNum { limbs: [ 0xcae90dfb3c0064c6123fa45fb28c2e, 0xd4cb52a34e3d7519670d09634336d4, 0x0023ce ] } , + BigNum { limbs: [ 0x61030f7a6df8cd67febb6c8cbe4070, 0xc60d84f924f255bef584e05d609204, 0x000a88 ] } , + BigNum { limbs: [ 0x5ca0f385906331980144927341bf91, 0xe145a4a45855dd7ae283294477734f, 0x006964 ] } , + BigNum { limbs: [ 0x192612437c0d705dd8aa1f184468b1, 0xac20308d7fa75d32092f3b381bf3aa, 0x002d1d ] } , + BigNum { limbs: [ 0xa47df0bc824e8ea22755dfe7bb9750, 0xfb32f90ffda0d607ced8ce69bc11a9, 0x0046cf ] } , + BigNum { limbs: [ 0xd10e7623a9b5c2385693e2fcd877bc, 0x930d0b536c1638fea19a2f6d48ccbf, 0x006c5c ] } , + BigNum { limbs: [ 0xec958cdc54a63cc7a96c1c03278845, 0x14461e4a1131fa3b366dda348f3893, 0x000791 ] } , + BigNum { limbs: [ 0x07f80a3c191747313bf2252bfe6544, 0xc3710f7d372068f5ddce3a6ac23aa6, 0x0048dd ] } , + BigNum { limbs: [ 0xb5abf8c3e544b7cec40dd9d4019abd, 0xe3e21a204627ca43fa39cf3715caad, 0x002b0f ] } , + BigNum { limbs: [ 0x7d31d54a5b5e0bf66850fc4356cc38, 0x5090296540c55c8ea24dbc4424135c, 0x000b63 ] } , + BigNum { limbs: [ 0x40722db5a2fdf30997af02bca933c9, 0x56c300383c82d6ab35ba4d5db3f1f7, 0x00688a ] } , + BigNum { limbs: [ 0x6754373db6852853633cba573000a7, 0xbbd2ed7a061996cedcf10a4bedd504, 0x0028da ] } , + BigNum { limbs: [ 0x564fcbc247d6d6ac9cc344a8cfff5a, 0xeb803c23772e9c6afb16ff55ea304f, 0x004b12 ] } , + BigNum { limbs: [ 0xefd2ccb27161336c0774388484c8e0, 0x40f3fe19d6697b3c0956d65c80f6ed, 0x005149 ] } , + BigNum { limbs: [ 0xcdd1364d8cfacb93f88bc67b7b3721, 0x665f2b83a6deb7fdceb13345570e65, 0x0022a4 ] } , + BigNum { limbs: [ 0xaf06cd4910e551088ec10c7b3f9602, 0xd80f951236ac65f6c38129c1631553, 0x004683 ] } , + BigNum { limbs: [ 0x0e9d35b6ed76adf7713ef284c069ff, 0xcf43948b469bcd431486dfe074f000, 0x002d69 ] } , + BigNum { limbs: [ 0x55cbf4fc7d9221d6816541aa07b714, 0x36935603e9159f28a666a615905a5d, 0x0043f2 ] } , + BigNum { limbs: [ 0x67d80e0380c9dd297e9abd55f848ed, 0x70bfd3999432941131a1638c47aaf6, 0x002ffb ] } , + BigNum { limbs: [ 0x3d713401b12579ff39b32afc7b0bc8, 0xbfc21f080b804db92c2e4bb771d4c2, 0x00566d ] } , + BigNum { limbs: [ 0x8032cefe4d368500c64cd40384f439, 0xe7910a9571c7e580abd9bdea663091, 0x001d7f ] } , + BigNum { limbs: [ 0xab8672e395976ddc3fce2d7273d9b8, 0xce96912a183ea6587ecd8746ffc65e, 0x000424 ] } , + BigNum { limbs: [ 0x121d901c68c49123c031d18d8c2649, 0xd8bc987365098ce1593a825ad83ef5, 0x006fc8 ] } , + BigNum { limbs: [ 0xd3cae6a6c85eb50b49338d697169d1, 0xfbbeff1e545584842b429aacfafecc, 0x003fd8 ] } , + BigNum { limbs: [ 0xe9d91c5935fd49f4b6cc71968e9630, 0xab942a7f28f2aeb5acc56ef4dd0686, 0x003414 ] } , + BigNum { limbs: [ 0x7cd05c252cdf03a7dac369758af536, 0xd634c2939a3328200b75d372ed58da, 0x0023af ] } , + BigNum { limbs: [ 0x40d3a6dad17cfb58253c958a750acb, 0xd11e6709e3150b19cc92362eeaac79, 0x00503d ] } , + BigNum { limbs: [ 0x981ce7693de7d08ae9eb8b6995ac0a, 0xa4253becc4b5eb2b624ae2bcf538a0, 0x0065ab ] } , + BigNum { limbs: [ 0x25871b96c0742e75161473966a53f7, 0x032dedb0b892480e75bd26e4e2ccb3, 0x000e42 ] } , + BigNum { limbs: [ 0x8d13ae231a5bdcc9fea91649dcfec3, 0xec6589730d57a091ebada7f04c5566, 0x00406b ] } , + BigNum { limbs: [ 0x309054dce40022360156e8b623013e, 0xbaeda02a6ff092a7ec5a61b18bafed, 0x003381 ] } , + BigNum { limbs: [ 0x13cf484f4959217912415ab836bd2c, 0x9d1f8742b93e0c418fd6b7d55b84cb, 0x00134b ] } , + BigNum { limbs: [ 0xa9d4bab0b502dd86edbea447c942d5, 0x0a33a25ac40a26f8483151cc7c8088, 0x0060a2 ] } , + BigNum { limbs: [ 0x97082543d2be0a3165acb7ba4029ee, 0xbbfa9954466c540e4ea7938b5f4ce7, 0x005448 ] } , + BigNum { limbs: [ 0x269bddbc2b9df4ce9a534745bfd613, 0xeb58904936dbdf2b8960761678b86c, 0x001fa4 ] } , + BigNum { limbs: [ 0xf8db8f62b9e688a40f1700385650df, 0xd3af97013194cff7b6d0a1aea46d99, 0x0047a0 ] } , + BigNum { limbs: [ 0xc4c8739d4475765bf0e8fec7a9af22, 0xd3a3929c4bb36342213767f33397b9, 0x002c4c ] } , + BigNum { limbs: [ 0x3129692575239a8f8cf7befeeb90de, 0x12cac8686c064244cec8b2e392eca6, 0x0068e9 ] } , + BigNum { limbs: [ 0x8c7a99da8938647073084001146f23, 0x948861351141f0f5093f56be4518ad, 0x000b04 ] } , + BigNum { limbs: [ 0x121bcca49e268a7218bcb84d7c3ebc, 0xff2e2a28d2d40b336334dc00d24146, 0x002455 ] } , + BigNum { limbs: [ 0xab88365b6035748de74346b283c145, 0xa824ff74aa74280674d32da105c40d, 0x004f97 ] } , + BigNum { limbs: [ 0x46d17864183203961a252dd8543362, 0x2856c609b4784ae25ffab9dc598650, 0x00391b ] } , + BigNum { limbs: [ 0x76d28a9be629fb69e5dad127abcc9f, 0x7efc6393c8cfe857780d4fc57e7f03, 0x003ad2 ] } +]; diff --git a/noir-projects/noir-protocol-circuits/crates/blob/src/smaller_config.nr b/noir-projects/noir-protocol-circuits/crates/blob/src/smaller_config.nr new file mode 100644 index 00000000000..40521353f01 --- /dev/null +++ b/noir-projects/noir-protocol-circuits/crates/blob/src/smaller_config.nr @@ -0,0 +1,36 @@ +use dep::bigint::{BigNum, fields::bls12_381Fr::BLS12_381_Fr_Params}; + +type F = BigNum<3, BLS12_381_Fr_Params>; + +global FIELDS_PER_BLOB: u64 = 8; // actually 4096 +global LOG_FIELDS_PER_BLOB: u64 = 3; // actually 12 + +// The following three fields are more meaningful computation in the proper config.nr: +global EXTRA_FIELDS_PER_BLOB: u64 = 0; // 0 = floor(8 FIELDS_PER_BLOB / 254 noir_field_bits). +global NOIR_FIELDS_PER_BLOB = FIELDS_PER_BLOB + EXTRA_FIELDS_PER_BLOB; // It's a more meaningful computation in the proper config.nr. +global FIELDS_CARRYING_AN_EXTRA_BIT_PER_BLOB: u64 = 0; // EXTRA_FIELDS_PER_BLOB * 254 = 0. It's a more meaningful computation in the proper config.nr. + +global D: F = BigNum { limbs: [8, 0, 0] }; +global D_INV = BigNum { limbs: [0x45ef829ffe907f1fffffff20000001, 0xf268c469cd9f2cd29d07086d9d04a9, 0x656f] }; + +global ROOTS: [F; FIELDS_PER_BLOB] = [ + BigNum { limbs: [ 0x000000000000000000000000000001, 0x000000000000000000000000000000, 0x000000 ] } , + BigNum { limbs: [ 0xbda402fffe5bfeffffffff00000000, 0xa753299d7d483339d80809a1d80553, 0x0073ed ] } , + BigNum { limbs: [ 0x030002760300000001000000000000, 0x0000000000008d51ccce760304d0ec, 0x000000 ] } , + BigNum { limbs: [ 0xbaa40089fb5bfefffeffff00000001, 0xa753299d7d47a5e80b39939ed33467, 0x0073ed ] } , + BigNum { limbs: [ 0x8b21c28713b7007228fd3397743f7a, 0x66f603fa66e78c0625cd70d77ce2b3, 0x003457 ] } , + BigNum { limbs: [ 0x32824078eaa4fe8dd702cb688bc087, 0x405d25a31660a733b23a98ca5b22a0, 0x003f96 ] } , + BigNum { limbs: [ 0x74903694b04fd86037fe81ae99502e, 0xb22e5ce11044babc5affca86bf658e, 0x001333 ] } , + BigNum { limbs: [ 0x4913cc6b4e0c269fc8017d5166afd3, 0xf524ccbc6d03787d7d083f1b189fc5, 0x0060b9 ] } +]; + +global NEGATIVE_ROOTS = [ + BigNum { limbs: [ 0xbda402fffe5bfeffffffff00000000, 0xa753299d7d483339d80809a1d80553, 0x0073ed ] } , + BigNum { limbs: [ 0x000000000000000000000000000001, 0x000000000000000000000000000000, 0x000000 ] } , + BigNum { limbs: [ 0xbaa40089fb5bfefffeffff00000001, 0xa753299d7d47a5e80b39939ed33467, 0x0073ed ] } , + BigNum { limbs: [ 0x030002760300000001000000000000, 0x0000000000008d51ccce760304d0ec, 0x000000 ] } , + BigNum { limbs: [ 0x32824078eaa4fe8dd702cb688bc087, 0x405d25a31660a733b23a98ca5b22a0, 0x003f96 ] } , + BigNum { limbs: [ 0x8b21c28713b7007228fd3397743f7a, 0x66f603fa66e78c0625cd70d77ce2b3, 0x003457 ] } , + BigNum { limbs: [ 0x4913cc6b4e0c269fc8017d5166afd3, 0xf524ccbc6d03787d7d083f1b189fc5, 0x0060b9 ] } , + BigNum { limbs: [ 0x74903694b04fd86037fe81ae99502e, 0xb22e5ce11044babc5affca86bf658e, 0x001333 ] } +];