Curve detail
Definition
| Name | sect571r1 (nist/B-571, secg/sect571r1, x962/ansit571r1) |
|---|---|
| Category | secg |
| Field | Binary |
| Field polynomial | $x^{571} + x^{10} + x^{5} + x^{2} + 1$ |
| Field bits | 571 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 |
| Param $b$ | 0x02f40e7e2221f295de297117b7f3d62f5c6a97ffcb8ceff1cd6ba8ce4a9a18ad84ffabbd8efa59332be7ad6756a66e294afd185a78ff12aa520e4de739baca0c7ffeff7f2955727a |
| Generator $x$ | 0x0303001d34b856296c16c0d40d3cd7750a93d1d2955fa80aa5f40fc8db7b2abdbde53950f4c0d293cdd711a35b67fb1499ae60038614f1394abfa3b4c850d927e1e7769c8eec2d19 |
| Generator $y$ | 0x037bf27342da639b6dccfffeb73d69d78c6c27a6009cbbca1980f8533921e8a684423e43bab08a576291af8f461bb2a8b3531d2f0485c19b16e2f1516e23dd3c1a4827af1b8ac15b |
| Simulation seed | 0x2aa058f73a0e33ab486b0f610410c53a7f132310 |
Characteristics
| Order | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe661ce18ff55987308059b186823851ec7dd9ca1161de93d5174d66e8382e9bb2fe84e47 |
| Cofactor | 0x2 |
| $j$-invariant | 0x45cbb32ced5555aa026b3b1728cec729c0d032e61381d4aae04bd3974ab437328b3d7123043021cc53b9e834f5cc04fde6ccbc0e9f4adb71d18678796b42848a95446bb29ef5928 |
| Trace $t$ | 0x333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6373 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe661ce18ff55987308059b186823851ec7dd9ca1161de93d5174d66e8382e9bb2fe84e47 |
| cofactor | 0x2 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x80000000000000000000000000000000000000000000000000000000000000000000000333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6374 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa411e0304016513267126336c9d6fdf330e270214f215b043cde1a9f73ceaa6b865a8feea8977c653bb8f06dad987e4074d9c34cf350d33e7d0771797bf42d62dbf3e2b6adc25aa |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x7', '0x864d', '0x169f44c4ecb', '0x18a70521054cbcd3a04031412f2e6ed0351922748c4f4ffed4c08b2b0dfc284fa3cd105d2667374d96863a2245268f472381f1080aaaa772c35baca966b316937'] |
| (+)largest_factor_bitlen | 0x201 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x5', '0x7', '0x15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15f15e8968fce0e667d694494352cf08134f49c4bfb2fea5365e9233e0060679a99477e23c311'] |
| (-)largest_factor_bitlen | 0x239 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x6ad', '0x9963e9d48f34474d71b1ce914d24ffd9a7058adc32ee2ca3938c5bacb6c009963e9d48ef6e11fd73fdd79f630495a8e150118b46d5400e8b0706ed2c686207ddc604276c9670b'] |
| (-)largest_factor_bitlen | 0x234 |
| $\text{torsion_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x6 |
| relative | 0x3 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xa |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x3 |
| full | 0xc |
| relative | 0x4 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x90 |
| full | 0x90 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6373 |
| factorization | ['0xb', '0x1f', '0x45d', '0x827', '0xa6c9', '0x52b0aa5e8edb', '0x2355df43f25e33', '0x38213de8f899295', '0xa9c9e966c86decb696039'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x2411e0304016513267126336c9d6fdf330e270214f215b043cde1a9f73ceaa6b865a8feea8977c653bb8f06dad987e4074d9c34cf350d33e7d0771797bf42d62dbf3e2b6adc25a9 |
| factorization | ['0x7', '0x2f', '0x512a20209af', '0x58862dbba935aa81f7ca4698801166d1b0d4cde364f1099dd1fdf4cf8b88e366d41778d49359c842351b5902107d3b18f46ce256db8704975378edbf138a25d9ef'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x12661ffce851db9cd65b8a7746d95206b3f8fcc1ffeedcd26d0cf118c4e83e7ce0e4706186849c6a8e41deeec39126a9094e65bd26c726bc9b3b53528113589cbaf8264e976636959105288747ea7b30987953b89746bd4e85373194350b18c6966129c6bf6dec65283ba15 |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | None |
| complement_bit_length | None |
| $\text{class_number}()$ | |
| upper | NO DATA (timed out) |
| lower | NO DATA (timed out) |
| $\text{small_prime_order}(l=2)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=3)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=5)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=7)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=11)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=13)$ | |
| order | None |
| complement_bit_length | None |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x1', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0xc', '0x1']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x2 |
| 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 | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=17)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=19)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0xc |
| relative | 0xc |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x9 |
| full | 0x9 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6373 |
| trace_factorization | ['0xb', '0x1f', '0x45d', '0x827', '0xa6c9', '0x52b0aa5e8edb', '0x2355df43f25e33', '0x38213de8f899295', '0xa9c9e966c86decb696039'] |
| number_of_factors | 0x9 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6373 |
| trace_factorization | NO DATA (timed out) |
| number_of_factors | NO DATA (timed out) |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x3 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x4 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x6 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x23a |
| expected | 0x11d |
| ratio | 0.5 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x27975 |
| expected | 0x13dd7 |
| ratio | 0.50175 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x1d47e88 |
| expected | 0xeb7bfe |
| ratio | 0.50264 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x333c63ce0154cf19eff4c9cf2fb8f5c27044c6bdd3c42d855d165322f8fa2c89a02f6372 |
| ratio | 7.765231125884475e+85 |
| distance 32 | 0xe |
| distance 64 | 0xe |
| $\text{multiples_x}(k=1)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=2)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=3)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=4)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=5)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=6)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=7)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=8)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=9)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=10)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{brainpool_overlap}()$ | |
| o | None |
| $\text{weierstrass}()$ | |
| a | None |
| b | None |