Curve detail
Definition
| Name | ed-510-mont |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x3eddffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) |
| Field bits | 510 |
| Form | Twisted Edwards $ax^2 + y^2 = 1 + dx^2y^2$ |
| Param $a$ | 0x3eddfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe |
| Param $d$ | 0x8da1e |
Characteristics
| Order | 0xfb77fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd7ced11e7c2f1abf716df42a6c246080b5fcc20917e59a42c85821cdf36d51b1 |
| Cofactor | 0x4 |
| $j$-invariant | 0x254391c7d08bafb2322725fd17ea39ef08eb3cb826dbb27f592956b10b874a5fa6b1783affd6e41c2db7740e5e52c82f24393bbc8c2be94770176e8c6e7c0199 |
| Trace $t$ | 0xa0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0xfb77fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd7ced11e7c2f1abf716df42a6c246080b5fcc20917e59a42c85821cdf36d51b1 |
| cofactor | 0x4 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x3ede000000000000000000000000000000000000000000000000000000000000a0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xf704883ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff697e819741b8fa3fab9db67c70e7f445d36dcd6e2f62a6923e7d43b9baaed374132bdec863be5fea0135b9285eb795f5b1232ee0db9fc82ad4d2f3eb0cb7c614 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2413c9', '0x1dc0ee48499d', '0xefe38c4591bba956c7b8e7c47ab53f04faa4c91e1e0cb24d3600767393b3db25dafab18c789a78f0d7d02422d917a52275b0e41a8d2ae67'] |
| (-)largest_factor_bitlen | 0x1bc |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x1325b', '0x46291cb402ad', '0x7fca02607762f8047388934d802871692aaef1b2346790c861785afda77976a0603a5ec722bc50f64984cb8a60d0f6db9de187265009c153'] |
| (-)largest_factor_bitlen | 0x1bf |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x1d', '0x3d', '0x1a3f', '0x1b7109', '0x1dd7e1', '0x9054d7b8a6415', '0x939a805eec2e9e0ad71ac8f4564e270b884cfd69b360f775a59c3ba3de67d96992207b12f59d08c3867fc505b452fd8fbf7'] |
| (+)largest_factor_bitlen | 0x18c |
| (-)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 | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)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 | ['0x7', '0xd', '0xd', '0x17', '0x12b9', '0x53e295', '0xa6d0c0f', '0x287c62ee18f42506a99', '0x77c0f6bb9d5e7e82fca5b9b9317217c0954ae7c8fc0dde741d24f8c7f8ee893a5d8b09b7ad64353cb26056bf00b'] |
| (+)largest_factor_bitlen | 0x16b |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x8 |
| full | 0x8 |
| 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 | 0x120 |
| full | 0x120 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0xa0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
| factorization | ['0x2', '0x2', '0x3', '0xb', '0xa7', '0x1ddf573222252d62c1042b2bad35772612de80761b1cad89e4b5703c3d2f9'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x2618819741b8fa3fab9db67c70e7f445d36dcd6e2f62a6923e7d43b9baaed374132bdec863be5fea0135b9285eb795f5b1232ee0db9fc82ad4d2f3eb0cb7c611 |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0xf8e785b7b4edc4d78b02dc3f74bef3c568cbdf3958852d14e454ac07a0a5887c20c42edced258b99464f6935075b464de27e652915ca4b7b63c0c7e0052affe7b91f0806ce77fb26c439cfb425eea70f3a9277c0f3112e38c4338d9719691c8 |
| 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', '0x1'], ['0x2', '0x1']] |
| len | 0x2 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0xc', '0x1']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x1 |
| $\text{volcano}(l=3)$ | |
| 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 | 0x2 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x12 |
| full | 0x12 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0xa0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
| trace_factorization | ['0x2', '0x2', '0x3', '0xb', '0xa7', '0x1ddf573222252d62c1042b2bad35772612de80761b1cad89e4b5703c3d2f9'] |
| number_of_factors | 0x5 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0xa0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
| trace_factorization | ['0x2', '0x7', '0x1b8c64e5', '0x10714562f83d1e5a029a6adbab5dc8267934522bcba4feab8ae8cc7de4b35028496ae9689bc2fe2aa973845f078afda91352527840d445294183313d'] |
| number_of_factors | 0x4 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x1 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x10d |
| expected | 0xff |
| ratio | 0.94796 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0xfdb8 |
| expected | 0xfd81 |
| ratio | 0.99915 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xa7bcfd |
| expected | 0xa7aefe |
| ratio | 0.99967 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x122000000000000000000000000000000000000000000000000000000000000a0c4bb860f4395023a482f564f6e7dfd280cf7dba06996f4de9f78c8324ab93c |
| ratio | 55.49655 |
| distance 32 | 0x4 |
| distance 64 | 0x4 |
| $\text{multiples_x}(k=1)$ | |
| 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 | 0x9d18bae4ea391b120b7d58874118615571e04e69ac87bb3f771e2f4b2cb41eef4714aad2e6cd634412cba01bca1919809594d1ba5a39cede888f00cc27caa07 |
| $\text{brainpool_overlap}()$ | |
| o | -0x109b5a6efcf8ac3da15ae173b596c616a2a902475c26a783fe14b4e11b1ae31b7d63c4114ad08bf13c3d7dc8 |
| $\text{weierstrass}()$ | |
| a | 0x21340df61e839336a7ed8a76d84c35f5728e8f5bde37462db6ce658e59c22b48479c87aaf51c3aef4fa9f439aea4e9985cfbbcecb33a7aa1f6da27fb8c9c89b |
| b | 0x2e7eced1d865929686f704283a108e9151fac5f6512146c798ff037aa0ea9eea48976bbb6a3e2e70f5074663ebb9417a331a9ca100e3ab3086c3efd3c82cb32b |