Curve detail
Definition
| Name | ed-256-mont |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0xffa7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) |
| Field bits | 256 |
| Form | Twisted Edwards $ax^2 + y^2 = 1 + dx^2y^2$ |
| Param $a$ | 0xffa7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe |
| Param $d$ | 0x350a |
Characteristics
| Order | 0x3fe9ffffffffffffffffffffffffffffb154bbafb86733c966d92eded8ec7bab |
| Cofactor | 0x4 |
| $j$-invariant | 0x518d79145010414d302bdda036cde5e5ba872b130a5dcf09e8ee5efbf18f09b8 |
| Trace $t$ | 0x13aad11411e6330da649b44849c4e1154 |
| Embedding degree $k$ | 0x3fe9ffffffffffffffffffffffffffffb154bbafb86733c966d92eded8ec7baa |
| CM discriminant | -0x9ef4bf2afb91f8dfa6e3fafa97647fec3e69b0aef222804896c8911cc3e8ef1b |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x3fe9ffffffffffffffffffffffffffffb154bbafb86733c966d92eded8ec7bab |
| cofactor | 0x4 |
| $\text{discriminant}()$ | |
| cm_disc | -0x9ef4bf2afb91f8dfa6e3fafa97647fec3e69b0aef222804896c8911cc3e8ef1b |
| factorization | ['0x2', '0x2', '0x71', '0xf43', '0x10fb5', '0xef03311f', '0x17cfd71b644129d26145a7299e39c167cf8d72c72b9747cb'] |
| max_conductor | 0x2 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0xffa800000000000000000000000000013aad11411e6330da649b44849c4e1154 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xff501e3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffd842d035411b81c8164701415a26e004f06593d443775fedda4ddbb8cf05c4394 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x13', '0xad', '0x1070b', '0x14dca63a3', '0x4f43846c4c829548fc244e9c7695d9391bc16fdcf1594cc6f'] |
| (-)largest_factor_bitlen | 0xc3 |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x53', '0x6290f6bf3a9a3784a062b2e43dafcea6148f0299c91a8761c6b4b7517fe1e6d'] |
| (-)largest_factor_bitlen | 0xfb |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x1d', '0x15091', '0x12422d97d', '0x1652d7a075', '0xca270e1c60fa63eec5550a3d988579d138c29739d9'] |
| (+)largest_factor_bitlen | 0xa8 |
| (-)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 | ['0x3', '0x5', '0x8909', '0x2a61f', '0x301484c8c7e98b6a312f457634a19ff737677e52eaddc6e0d1a70547'] |
| (-)largest_factor_bitlen | 0xde |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x3', '0x2f', '0x44153', '0x9af054d', '0x99d6c800b', '0x47a6c98b289b9', '0x194d86e32302b49', '0xd3c0aaa423bf2f45'] |
| (+)largest_factor_bitlen | 0x40 |
| (-)factorization | ['0xff70f7', '0x50113c715e73bef9bf389856157100e013dce3b19ab7abc528bb29915bd'] |
| (-)largest_factor_bitlen | 0xeb |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x5', '0x5', '0x17', '0x67', '0x137', '0x12631b22d765', '0xde4940a09e4e1e0e02b', '0x57837cd1e173e3542f815dc950721'] |
| (+)largest_factor_bitlen | 0x73 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x11', '0x133', '0x338b9', '0x773839d3', '0x3a829a699db55a7d3281cf84b609ea138e5520e66a01605fe5'] |
| (+)largest_factor_bitlen | 0xc6 |
| (-)factorization | ['0x3', '0x3', '0x19f85', '0x109093', '0x11d7a77', '0x6a1c8bde0f6c4a98a4164d0f06aa4862ba357b9a733950d63'] |
| (-)largest_factor_bitlen | 0xc3 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x25', '0x71', '0x191', '0x1aa623563a6bdef227491259af97ed82f779d1bed6a51503190a83e4b02f'] |
| (+)largest_factor_bitlen | 0xed |
| (-)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 | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x28 |
| full | 0x28 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x38 |
| full | 0x38 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x8 |
| full | 0x10 |
| relative | 0x2 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x13aad11411e6330da649b44849c4e1154 |
| factorization | ['0x2', '0x2', '0x7', '0x3b3', '0x827', '0xdb67', '0x6f50847245f6adc29d4501'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x8325035411b81c8164701415a26e004f06593d443775fedda4ddbb8cf05c4391 |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x990cd6d401455e1ef1fd7bfb96475befef842246518172ef6ef18bea57910adc02b216538249ac5fc375053972c12218 |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x1 |
| complement_bit_length | 0x1 |
| $\text{class_number}()$ | |
| upper | 0x2c6357fbe3accfe956e9dd9c119b371ab0 |
| lower | 0xd54034 |
| $\text{small_prime_order}(l=2)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x3fe9ffffffffffffffffffffffffffffb154bbafb86733c966d92eded8ec7baa |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x1ff4ffffffffffffffffffffffffffffd8aa5dd7dc3399e4b36c976f6c763dd5 |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x3fe9ffffffffffffffffffffffffffffb154bbafb86733c966d92eded8ec7baa |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x442ccccccccccccccccccccccccccccc78e2ea5508f69da3a0e7a9763cb7fb6 |
| complement_bit_length | 0x5 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x1ff4ffffffffffffffffffffffffffffd8aa5dd7dc3399e4b36c976f6c763dd5 |
| complement_bit_length | 0x3 |
| $\text{division_polynomials}(l=2)$ | |
| 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 | 0x2 |
| 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 | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0xe |
| full | 0xe |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x10 |
| relative | 0x10 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0xa |
| full | 0xa |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x13aad11411e6330da649b44849c4e1154 |
| trace_factorization | ['0x2', '0x2', '0x7', '0x3b3', '0x827', '0xdb67', '0x6f50847245f6adc29d4501'] |
| number_of_factors | 0x6 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x13aad11411e6330da649b44849c4e1154 |
| trace_factorization | ['0x2', '0x3', '0x3', '0xb', '0x161', '0x6a1', '0x119caac9e48600cdcd855ac5ed5f0bbb71fa915bbfd9f4cc857b2ebd1d'] |
| number_of_factors | 0x6 |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0x77 |
| expected | 0x80 |
| ratio | 1.07563 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x3fe4 |
| expected | 0x3fc0 |
| ratio | 0.9978 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x15205b |
| expected | 0x151580 |
| ratio | 0.99799 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x5800000000000000000000000000013aad11411e6330da649b44849c4e1154 |
| ratio | 743.72727 |
| 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 | 0xefe0c841e3ef76e32aa751d4ce5413a0dc21292f1a5a8c984de2a6d9ad6b03ff |
| $\text{brainpool_overlap}()$ | |
| o | -0xa5cc80f39ec7949ba31f74d1 |
| $\text{weierstrass}()$ | |
| a | 0x46f46e898d76fdeb3e1c847d22da305ea66abef15087e591602e22654cdc0183 |
| b | 0xf6546684fef5b700effb7654871f4eb20bf3c09a23f76395492eec0c94ad9ee2 |