Curve detail
Definition
| Name | w-510-mont |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) |
| Field bits | 510 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x3eddfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc |
| Param $b$ | 0x988d |
Characteristics
| Order | 0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb9146ccde150ef33747ab29d1e6573d8d22de95e322303f3a00b200986fa9a2d |
| Cofactor | 0x1 |
| $j$-invariant | 0x243063c1052f01002754a0b67ab8c46f9cf001ca12e81aa08295593eb3f9b4524ead8ef2416ed9ac4b0ac69dbecdceefec505eee0f153139805293078f617e02 |
| Trace $t$ | 0x46eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb9146ccde150ef33747ab29d1e6573d8d22de95e322303f3a00b200986fa9a2d |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x3ede00000000000000000000000000000000000000000000000000000000000046eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xf704883ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff182dad46fb6eaba5c13004d7a750b35ca68b2565c2f67e928405f3fe5f375b8f9f6a2436cdf1dfcf0c5423770d65cad74bc2e3938938fbc8eb3365e298be2bed |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2', '0x2', '0x97', '0x1b7', '0x89b', '0x1def551f940309f7', '0xf7107e219703f04ec53aa1b28a686f916ca7fabeb0f02b46f851c67758349c270e28e42d9559ebe83293e89ac9d21d4d826a272e7'] |
| (-)largest_factor_bitlen | 0x1a4 |
| $\text{kn_factorization}(k=2)$ | |
| (+)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 | ['0x5', '0x7', '0xa7', '0x10d', '0x265', '0xa67', '0x4086badf', '0x6edc9e23', '0x3dae882a9e6d5cd935895f765912cad759474f1571970a9f53c7b94ca9557d7abe6cc3a0fa813f54509eb5a1ab219fd0fd44135'] |
| (-)largest_factor_bitlen | 0x19a |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2', '0x2', '0x2', '0x2', '0x2', '0x3', '0xd', '0x105cb0f', '0x5c4eef8e15', '0xaedd253bb6ceacdfea92803d55ba081d65865586a01008863439277b9ec3b25729c71a7ca7f490c61c030274133c13fdfccba84c2331a3'] |
| (-)largest_factor_bitlen | 0x1b8 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x5', '0x61', '0x1994f5', '0x10f9777', '0x7560a3c9230cd9808ca3c0ee93a93d8c202890af63a05a94f091ea2aa4c744f363e17254296d4c9670d95daad09238e8c3014b0e72c8070e101'] |
| (+)largest_factor_bitlen | 0x1cb |
| (-)factorization | ['0x4d35c7', '0xf006a0f6f7', '0x535e6548daf8b06773ef1851e640c59035bdc78b4ade3335b8c32b52a3bede8d007eba5c32fb4fc0c891f94aedc08adb0e8fceaabdeae240d'] |
| (-)largest_factor_bitlen | 0x1c3 |
| $\text{kn_factorization}(k=7)$ | |
| (+)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 | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xa |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x60 |
| full | 0x60 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x46eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
| factorization | ['0x3', '0x5', '0x5', '0x7', '0x1f', '0x2f', '0x7c285', '0x797fbd9', '0x3294b163e12e2f01', '0x859aa142207792fa6e332557b2caccbb'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x2b3852b90491545a3ecffb2858af4ca35974da9a3d09816d7bfa0c01a0c8a4706095dbc9320e2030f3abdc88f29a3528b43d1c6c76c7043714cc9a1d6741d416 |
| factorization | ['0x2', '0xd', '0x17f', '0x10e2ff', '0x1de2bf', '0x79931e58f', '0x123d497d6f2ee2d3', '0x18409fad2bb831623', '0xafd4aa3ee2e0b4c97ca17af51ceb1cc05d9c0ab51df3c8c976a22d04eed46fad595f5928ff'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x1d63ba1c1daa7e25d7a0d2d35ec7ffd892ffe92a275402fb0a6a1a632cf6467a0d2dab9cb768770536866d3e165fd5e184c831adb894ca25675d23163056d72d08e4f39d2902287043806c65134dca601314310543b47c1345d5d8ae2557164f |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x4 |
| complement_bit_length | 0x3 |
| $\text{class_number}()$ | |
| upper | NO DATA (timed out) |
| lower | NO DATA (timed out) |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb9146ccde150ef33747ab29d1e6573d8d22de95e322303f3a00b200986fa9a2c |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb9146ccde150ef33747ab29d1e6573d8d22de95e322303f3a00b200986fa9a2c |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0xfb77fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffee451b3378543bccdd1eaca747995cf6348b7a578c88c0fce802c80261bea68b |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0xfb77fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffee451b3378543bccdd1eaca747995cf6348b7a578c88c0fce802c80261bea68b |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x1f6effffffffffffffffffffffffffffffffffffffffffffffffffffffffffffdc8a3666f0a87799ba3d594e8f32b9ec6916f4af191181f9d0059004c37d4d16 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffb9146ccde150ef33747ab29d1e6573d8d22de95e322303f3a00b200986fa9a2c |
| complement_bit_length | 0x1 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x4', '0x3']] |
| 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 | 0x0 |
| 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 | 0x0 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0xa |
| relative | 0xa |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x5 |
| full | 0x5 |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x46eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
| trace_factorization | ['0x3', '0x5', '0x5', '0x7', '0x1f', '0x2f', '0x7c285', '0x797fbd9', '0x3294b163e12e2f01', '0x859aa142207792fa6e332557b2caccbb'] |
| number_of_factors | 0x9 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x46eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
| trace_factorization | NO DATA (timed out) |
| number_of_factors | NO DATA (timed out) |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0xf6 |
| expected | 0xff |
| ratio | 1.03659 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0xfddf |
| expected | 0xfd81 |
| ratio | 0.99855 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xa7a6f7 |
| expected | 0xa7aefe |
| ratio | 1.00019 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x12200000000000000000000000000000000000000000000000000000000000046eb93321eaf10cc8b854d62e19a8c272dd216a1cddcfc0c5ff4dff6790565d3 |
| ratio | 55.49655 |
| distance 32 | 0xd |
| distance 64 | 0x13 |
| $\text{multiples_x}(k=1)$ | |
| 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 |
| $\text{x962_invariant}()$ | |
| r | 0x330454f8c6da76fa882c6c107d31d48abe8f1875ffff40ca3698abe9956449fbf41ee7f9781d4a880b09ab90656dd47280ee93bfcaf97e4f2c610c822e2a984b |
| $\text{brainpool_overlap}()$ | |
| o | 0x1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc |
| $\text{weierstrass}()$ | |
| a | 0x3eddfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc |
| b | 0x988d |