Curve detail
Definition
| Name | id-GostR3410-2001-CryptoPro-A-ParamSet |
|---|---|
| Category | gost |
| Description | RFC4357 |
| Field | Prime (0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd97) |
| Field bits | 256 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd94 |
| Param $b$ | 0xa6 |
| Generator $x$ | 0x01 |
| Generator $y$ | 0x8d91e471e0989cda27df505a453f2b7635294f2ddf23e3b122acc99c9e9f1e14 |
Characteristics
| Order | 0xffffffffffffffffffffffffffffffff6c611070995ad10045841b09b761b893 |
| Cofactor | 0x1 |
| $j$-invariant | 0x5f71484039164cb5f71484039164cb5f71484039164cb5f71484039164cb5e8b |
| Trace $t$ | 0x939eef8f66a52effba7be4f6489e4505 |
| Embedding degree $k$ | 0x7fffffffffffffffffffffffffffffffb63088384cad688022c20d84dbb0dc49 |
| CM discriminant | -0x3aae01634baee862408b77a2f75c8edff71d8e9fbcc7b515ed97b09b7eb384443 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0xffffffffffffffffffffffffffffffff6c611070995ad10045841b09b761b893 |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | -0x3aae01634baee862408b77a2f75c8edff71d8e9fbcc7b515ed97b09b7eb384443 |
| factorization | ['0x17', '0x71', '0x1e8bd', '0x521fa5b908b', '0x61156ac8fa59f1af7', '0x18e2c1419fc3a160b517b0a793168bd'] |
| max_conductor | 0x1 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x100000000000000000000000000000000939eef8f66a52effba7be4f6489e429d |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb2c551fe9cb451179dbf74885d08a3712008e2716043384aea12684f64814cd85fd |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x2', '0x2', '0x11b', '0x39e4dcb8897f8c36468eed00e79372e2049bd4add1447b40499d6c556b81bf'] |
| (+)largest_factor_bitlen | 0xf6 |
| (-)factorization | ['0x2', '0x3', '0x7', '0x11', '0x25', '0x7f', '0x1da95', '0x14d5ce217', '0x2124870741dadb3093ea3747085493f1e0d4be7ba1b1425'] |
| (-)largest_factor_bitlen | 0xba |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x3', '0x89', '0x2a011', '0x117e0273b', '0x87e0cc8e9324d', '0x3f6fc5ad63ffe351b', '0x34cd0742691f988d9c031'] |
| (+)largest_factor_bitlen | 0x52 |
| (-)factorization | ['0x5', '0x1d', '0x2e7', '0x2db5', '0x12907', '0x44b55', '0x15e1c2fbe66c9e2fe1b4748539b0b811225e3277efe445b8d'] |
| (-)largest_factor_bitlen | 0xc1 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x5', '0x359', '0x24827', '0x10c09d16cbf45', '0x2a1f7a91283315', '0x3a5b932c201cbdc941466948e24e869f'] |
| (+)largest_factor_bitlen | 0x7e |
| (-)factorization | ['0x2', '0x2', '0x2', '0x17', '0x3d', '0x54fad', '0xaa05563ef', '0x4f743afb65131d2082b86372fd971955e494a38793f468a7'] |
| (-)largest_factor_bitlen | 0xbf |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0xb', '0xa7f', '0x2ff9743c7', '0xb79f25aa81e50b605', '0x41fb6714652a9d3a2f85fec8403e685127f3'] |
| (+)largest_factor_bitlen | 0x8f |
| (-)factorization | ['0x3', '0x3', '0x132d', '0x153d', '0x7a7d', '0x38afcc79e0905', '0x85abef7cde90ea7ca67', '0x50ccba5b563479b2356054d'] |
| (-)largest_factor_bitlen | 0x5b |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x2', '0x2', '0x2', '0x2', '0x2', '0x3', '0x3', '0x3', '0x19479c1', '0x415c8399a37', '0xbd07b51d23ae5a9693781d', '0x4f93e64d82cde5a93212914f'] |
| (+)largest_factor_bitlen | 0x5f |
| (-)factorization | ['0x2', '0x27ffffffffffffffffffffffffffffffe8ef2a9197f630a80adca43984a744d6f'] |
| (-)largest_factor_bitlen | 0x102 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x5d075e608e1a9', '0x1082d22101f9ce469faab68e9dae0c3f24ebee32dd8c33b571089'] |
| (-)largest_factor_bitlen | 0xd1 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0x13', '0xad', '0x24b', '0x7a3ff3', '0x2d6a21d', '0x3a79023', '0x2c1b1b03', '0x23a8de75f4e46742950391273567ee8361'] |
| (+)largest_factor_bitlen | 0x86 |
| (-)factorization | ['0x2', '0x2', '0x3', '0x5', '0xb', '0xd', '0xa3', '0x86e9', '0xb76d9', '0x197763ab34f8d', '0x8bb8e9c8983761365bb2084b8e079895bdf24d7'] |
| (-)largest_factor_bitlen | 0x9c |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x5', '0x49', '0xf59', '0xb999', '0x1d0b7', '0xeeac2a5', '0x2ff4602cf', '0x87bb5e039461695a22a830ae8f6cbca0debb'] |
| (+)largest_factor_bitlen | 0x90 |
| (-)factorization | ['0x7', '0x1f', '0xf551', '0x9476dd', '0x10fb84c887adca947356a2309486eacc37311d45d0e5183742a7cb'] |
| (-)largest_factor_bitlen | 0xd5 |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x78 |
| full | 0x78 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x54 |
| full | 0x54 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x939eef8f66a52effba7be4f6489e4505 |
| factorization | ['0x13', '0x50b', '0x53037d', '0x235d4547', '0x4d6aa5c1', '0x71bb209f'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0xaae01634baee862408b77a2f75c8edff71d8e9fbcc7b515ed97b09b7eb384b7e |
| factorization | ['0x2', '0x5', '0x7', '0x3d', '0x1aad', '0x12695b', '0x556fc63950db984058e3b56be949ff05d5d4763ca6e35a55956a7'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0xf627b2844ef2b1dc9857d38def3bcd683b800ca88c5e24f5049504b52fc1a394a3f911a2e0ef9b6daaf4b276c6c21669 |
| factorization | ['0x3', '0x13', '0x3b', '0x50b', '0x53037d', '0x235d4547', '0x4d6aa5c1', '0x71bb209f', '0x26966d091b05a6e1fde5321889ec8f016568f0fe2c84a38f151b081c32307a5'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x2 |
| complement_bit_length | 0x2 |
| $\text{class_number}()$ | |
| upper | 0x6cf5d8fca366c982fa4a6af0efe14369e8 |
| lower | 0x644016d6 |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x249249249249249249249249249249247d32701015e8670009ee4d0163571a5e |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0xc30c30c30c30c30c30c30c30c30c30c29bb7ab0074d77aaadfa19ab211d08ca |
| complement_bit_length | 0x5 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0xffffffffffffffffffffffffffffffff6c611070995ad10045841b09b761b892 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0xc30c30c30c30c30c30c30c30c30c30c29bb7ab0074d77aaadfa19ab211d08ca |
| complement_bit_length | 0x5 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x618618618618618618618618618618614ddbd5803a6bbd556fd0cd5908e8465 |
| complement_bit_length | 0x6 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x5555555555555555555555555555555524205ad0331e45aac1d6b3ade7cb3d86 |
| complement_bit_length | 0x2 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x6', '0x2']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=17)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=19)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x7 |
| full | 0x7 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x8 |
| relative | 0x8 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x939eef8f66a52effba7be4f6489e4505 |
| trace_factorization | ['0x13', '0x50b', '0x53037d', '0x235d4547', '0x4d6aa5c1', '0x71bb209f'] |
| number_of_factors | 0x6 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x939eef8f66a52effba7be4f6489e4505 |
| trace_factorization | ['0x3', '0x3b', '0x26966d091b05a6e1fde5321889ec8f016568f0fe2c84a38f151b081c32307a5'] |
| number_of_factors | 0x3 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x8a |
| expected | 0x80 |
| ratio | 0.92754 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x402a |
| expected | 0x3fc0 |
| ratio | 0.99355 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x151a65 |
| expected | 0x151580 |
| ratio | 0.99909 |
| $\text{square_4p1}()$ | |
| p | 0x5 |
| order | 0x3 |
| $\text{pow_distance}()$ | |
| distance | 0x939eef8f66a52effba7be4f6489e476d |
| ratio | 5.9010830870259516e+38 |
| distance 32 | 0xd |
| distance 64 | 0x13 |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0x1 |
| bits | 0x1 |
| difference | 0xff |
| ratio | 0.00391 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0x5af72a60c16a413e0d46b8a3aa1f32ed8c3e6b9dfe4648ecc7396b29b743a1b4 |
| bits | 0xff |
| difference | 0x1 |
| ratio | 0.99609 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0x4c5bdcd7c71c3de9332fa21da742799f0b13771d8ef4b31165ccd012f3319a75 |
| bits | 0xff |
| difference | 0x1 |
| ratio | 0.99609 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0x8ea4cc2b0080a3f2d23e1495882cf9c3834ccd0a0442f76ee8652f58125a8c82 |
| bits | 0x100 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0x5ff6395fee0840782f8ddc539973d01fe301e89d924fb663d9f710a90f45b62f |
| bits | 0xff |
| difference | 0x1 |
| ratio | 0.99609 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0xd8747558339a831395acee6ab13d38bf3587cb25f3cd320446e35671ff954e8f |
| bits | 0x100 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0xf7772fece0d8fc3e8da57a52f5bde640cdcde191181c9259556c591d484c2058 |
| bits | 0x100 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0x1dc96b8aa13666193e3c4e5b3ba67551638a6b54f41a1841fdbce9e52d593aa3 |
| bits | 0xfd |
| difference | 0x3 |
| ratio | 0.98828 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0x4aec54430d20cec901b953bd9bfb8e3dd7541c0bebe55d7fbbd0108e69154242 |
| bits | 0xff |
| difference | 0x1 |
| ratio | 0.99609 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0xd30dedd81aad7dc57c5ac95a8b6064e45ca6ddac00838cad75ccafe9894b8575 |
| bits | 0x100 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{x962_invariant}()$ | |
| r | 0x73ba495aeea55d758a0d4db3f81f3708f4aa7e45a39e5779ec9a247de681fbdd |
| $\text{brainpool_overlap}()$ | |
| o | 0x7ffffffffffffffffffffd94 |
| $\text{weierstrass}()$ | |
| a | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd94 |
| b | 0xa6 |