Curve detail
Definition
| Name | ed-511-mers |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1f) |
| Field bits | 511 |
| Form | Twisted Edwards $ax^2 + y^2 = 1 + dx^2y^2$ |
| Param $a$ | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1e |
| Param $d$ | 0x10bf7d |
Characteristics
| Order | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffea7c34182e41e2e9baa930e478c489b72542706bec5f32194f7c2e8f8d142f11 |
| Cofactor | 0x4 |
| $j$-invariant | 0x6b981e37d43b3b5947e5fc7ebb6f6777a4cdb7c4fdd5ed19ac8f788ce94c57cdd5643c34cd35172acffa8dfbddf1881922a63699700fc6554b1fe7dce73c0d75 |
| Trace $t$ | 0x560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf41dc |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffea7c34182e41e2e9baa930e478c489b72542706bec5f32194f7c2e8f8d142f11 |
| cofactor | 0x4 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x8000000000000000000000000000000000000000000000000000000000000000560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf3ffc |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1e1cee34e5a1351e71f3cfda8d7104d1565a58bcc85d7e9fb2ee556c2141b2338b88981b1abc7a4ab05cdc5b406018d0610e295450ccbd4d8c6199502603bd0094 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0xa7', '0xe5640575', '0x1b5f3fe351bb7f46628a97d9c529d2ffa6ad556f06d91e13b8a2a7ad91bc2d20be5c9049c0579067dd9aa85e3e6dc57b0d21b72e8815d71549a0933'] |
| (+)largest_factor_bitlen | 0x1d9 |
| (-)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 | ['0x3', '0x29', '0x4dcf', '0x67cc91f', '0x21c6f670e5a789ed3662ade0faf55d27351ee6d531b3456ee0c063f7ab6e851353da56d7e524357110e27e42fca80ccce5e0a31210f6c126ed8b3'] |
| (+)largest_factor_bitlen | 0x1d2 |
| (-)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 | ['0x3', '0xb', '0x13', '0x45fa71b59a71d73a9', '0x3bbefb6c54f6a57d878bba5b9892717ab4fffbe3f4fe315e07cd6f919c20754667548ec6eeefb537db6271885c403cc57e8a6145b2e939'] |
| (-)largest_factor_bitlen | 0x1b6 |
| $\text{kn_factorization}(k=6)$ | |
| (+)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 | 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 | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xb |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0xa8 |
| full | 0xa8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x48 |
| full | 0x48 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf41dc |
| factorization | ['0x2', '0x2', '0x3', '0x3', '0x3', '0x3', '0x5', '0x29', '0x47', '0x1df', '0x50b', '0x1ebb', '0x394d', '0x17a0b', '0x331c37a1c6f2e813094f3e94891f5822f5c6776828d'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x6311cb1a5ecae18e0c3025728efb2ea9a5a74337a281604d11aa93debe4dcc747767e4e54385b54fa323a4bf9fe72f9ef1d6abaf3342b2739e66afd9fc46890f |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x4c5572834dadf06a1a764ec985428209d2ba7ddd7bda2de7ca9ee655197e5cd0610349cc49bb0d1ed099ea52cdd61c9ba86c85f93604dc66844fc4697a6a88c17c0441b36599624462d962277eba68e3f0a156cd5a0b2ecd3f4229789760d988 |
| 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 | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x4', '0x3']] |
| 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 | 0x1 |
| 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 | 0x0 |
| 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 | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0xb |
| relative | 0xb |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0xe |
| full | 0xe |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x9 |
| full | 0x9 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x14 |
| full | 0x14 |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf41dc |
| trace_factorization | ['0x2', '0x2', '0x3', '0x3', '0x3', '0x3', '0x5', '0x29', '0x47', '0x1df', '0x50b', '0x1ebb', '0x394d', '0x17a0b', '0x331c37a1c6f2e813094f3e94891f5822f5c6776828d'] |
| number_of_factors | 0xb |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf41dc |
| trace_factorization | ['0x2', '0x7', '0xfa82c3984f36b', '0x10931c1559271e6cb84166751a7f10d42b3c7ea4621812a15b8afe4787db3fa688166aed80264a0a644b602681df3de36577e62195836306113'] |
| 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 | 0x10f |
| expected | 0xff |
| ratio | 0.94096 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0xfed5 |
| expected | 0xfe80 |
| ratio | 0.9987 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xa89cc8 |
| expected | 0xa8ac7f |
| ratio | 1.00036 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x560f2f9f46f87459155b3c6e1cedd9236af63e504e83379ac20f45c1cbaf43bc |
| ratio | 1.722229196364211e+77 |
| 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 | 0x1b7bdedd100dd57c7a98a4c4ff1a81c5ff9783d7d784a420a872240367bd896aafcfd3c8eee9d0d3c2d73e516566d916195a25abb073ca5901a5b41d9b455e2a |
| $\text{brainpool_overlap}()$ | |
| o | 0x127f490468e5273795bf48e3e3db704f04dd93d954585c6d4850e60aa1746fa515fcf9ab180e4dc6e6512bbd |
| $\text{weierstrass}()$ | |
| a | 0xfd40c57f370da3593e1bf4157c0f8cd948c16e0694bdc642d1e95b75d22517b37f21d69a880dd44dba6cee21664cfe7ae785e303a97c5a07f600846b5e26731 |
| b | 0x16cc935fc4396e7fc76308975416ad1aa3a3496b874e7274ce13e9dd0d03ee8b249acbf56751ba7fcf913b748e77587570bedf75f591618e12d42638efd18635 |