Curve detail
Definition
| Name | ed-255-mers |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd03) |
| Field bits | 255 |
| Form | Twisted Edwards $ax^2 + y^2 = 1 + dx^2y^2$ |
| Param $a$ | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd02 |
| Param $d$ | 0xea97 |
Characteristics
| Order | 0x1fffffffffffffffffffffffffffffffdcf1a785eda6832eac49d1ed0436eb75 |
| Cofactor | 0x4 |
| $j$-invariant | 0x5f27924db0ae3c434896c8273161414babc5f329356c0303474178e595c90c31 |
| Trace $t$ | 0x8c3961e84965f3454ed8b84bef244f30 |
| Embedding degree $k$ | 0x1fffffffffffffffffffffffffffffffdcf1a785eda6832eac49d1ed0436eb74 |
| CM discriminant | -0x6ccc4c034c357d085f2dc64d1654eff361611a1f206b5113bb9ff237808052c3 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x1fffffffffffffffffffffffffffffffdcf1a785eda6832eac49d1ed0436eb75 |
| cofactor | 0x4 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x800000000000000000000000000000008c3961e84965f3454ed8b84bef244c34 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd024ccecff2cf2a0bde8348e6cba6ac40327a7b97837e52bbb111803721fe079d04 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x7', '0x1d', '0x91b64ef2a7', '0x11b983ea795c5f89ce89e5a4322422baab4344ec38d08416fb009'] |
| (+)largest_factor_bitlen | 0xd1 |
| (-)factorization | ['0x3', '0x7c3', '0xceaa8599', '0x1b2c5f6bec7ebadc783984b', '0x40267959be22330b2cb0146bc43a0f9'] |
| (-)largest_factor_bitlen | 0x7b |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x3', '0x5', '0x5', '0xd', '0x191', '0x1af', '0x593', '0x298f7', '0x148a3d95', '0x15f08f16bbf67f6f281be0e38541e36ab51f21a8d321'] |
| (+)largest_factor_bitlen | 0xad |
| (-)factorization | ['0xb', '0x2b', '0xbea17', '0xba10cbaa705a56c1eac861a39b1bc061f88b273f722b715119c0df4d9'] |
| (-)largest_factor_bitlen | 0xe4 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x15cd8e870466e35', '0x24c2200134792eb', '0x7aa8c60b37c355638fa4d144631ee301513b'] |
| (+)largest_factor_bitlen | 0x8f |
| (-)factorization | ['0x5', '0x7f', '0x9acf3809acf3809acf3809acf3809ace8e71a0f7fe4f13d47d90aa93a1e501'] |
| (-)largest_factor_bitlen | 0xf8 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x3', '0x85fd', '0x6cb1587465199d0493a1a8faa7c9980a5e4a621d8c6c42b97c011a84e523'] |
| (-)largest_factor_bitlen | 0xef |
| $\text{kn_factorization}(k=5)$ | |
| (+)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 | ['0x2a5', '0x6fb', '0x1485d', '0x11463b', '0x1e0ace4d135f9c04b3212dce9b1ef28f19689d9fd6454b19f31'] |
| (+)largest_factor_bitlen | 0xc9 |
| (-)factorization | ['0x7', '0x4a9315e3ad6dc15', '0x178a0be019a86dd29357238f58e9209d04290ffcc6fc04fa8d'] |
| (-)largest_factor_bitlen | 0xc5 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x5', '0x95', '0x95', '0x210fccbf3898b51eb2a6fb9eda80e269c5ee902ce7f7365e33de858effcf1'] |
| (+)largest_factor_bitlen | 0xf2 |
| (-)factorization | ['0x3', '0xa2d966073de7b5c4ae75', '0x1d58182ca248dd21ef20007546bcf5ac20bcb61486e15'] |
| (-)largest_factor_bitlen | 0xb1 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x7', '0x25', '0x61', '0x161', '0x142e1', '0x4b4ff4caa9', '0x106829421a8aa3', '0x6a126bce61a6db2fa306ca9f307825bb'] |
| (+)largest_factor_bitlen | 0x7f |
| (-)factorization | ['0x5', '0x11', '0x11', '0x8dd', '0x95b7', '0x10021d', '0x22fb0fe673bc33f32bb58278b40891f9d48939079d381278abd'] |
| (-)largest_factor_bitlen | 0xca |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x3c |
| full | 0x3c |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x54 |
| full | 0x54 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x90 |
| full | 0x90 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x8c3961e84965f3454ed8b84bef244f30 |
| factorization | ['0x2', '0x2', '0x2', '0x2', '0x14a0a1', '0x1fd332d0f9', '0x36aebba891ecda96b'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x3331300d30d5f4217cb719345953bfcd8584687c81ad444eee7fc8de02015403 |
| factorization | ['0x20e3', '0x1bf23', '0x8a0f6577a5', '0x1a70d5e704f74ddee7c6d8bc129123887a781266b55680f'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x622710c324cdcffeeb3f443cd383a8bd32a7be4581c231c2113a40a564084681cf44a9de63b30bcb63b8bbd4c7180b20 |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x1 |
| complement_bit_length | 0x1 |
| $\text{class_number}()$ | |
| 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 | [['0x4', '0x1']] |
| 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 | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x0 |
| 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 | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x7 |
| full | 0x7 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x9 |
| full | 0x9 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x8c3961e84965f3454ed8b84bef244f30 |
| trace_factorization | ['0x2', '0x2', '0x2', '0x2', '0x14a0a1', '0x1fd332d0f9', '0x36aebba891ecda96b'] |
| number_of_factors | 0x4 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x8c3961e84965f3454ed8b84bef244f30 |
| trace_factorization | ['0x2', '0x3', '0x8b', '0x217f49', '0x23dd2dd7c9b', '0x20a4e94f9ccd522ab196d', '0x5beaf8674f48155804a9405975'] |
| number_of_factors | 0x7 |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0x8a |
| expected | 0x7f |
| ratio | 0.92029 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x3f31 |
| expected | 0x3f40 |
| ratio | 1.00093 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x14da50 |
| expected | 0x14d63f |
| ratio | 0.99924 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x8c3961e84965f3454ed8b84bef24522c |
| ratio | 3.106179848792591e+38 |
| distance 32 | 0xc |
| distance 64 | 0x14 |
| $\text{multiples_x}(k=1)$ | |
| 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 | 0x6d2b5d6360297bb13425013cbf1286500b031abdb4129ecc2938238b1c51909b |
| $\text{brainpool_overlap}()$ | |
| o | -0x369356bece4e3b8c063269db |
| $\text{weierstrass}()$ | |
| a | 0x81c37095ee5056370f1702ce15f56dd94742a80137215bab3b6bf92a4baa236 |
| b | 0x4a056c798204fb1eaaed0c11cf237bad7172514b4d2cc4dfd966cb2d64d2405b |