Curve detail
Definition
| Name | w-511-mers |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1f) |
| Field bits | 511 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1c |
| Param $b$ | 0x879da |
Characteristics
| Order | 0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff8dbefa3f5ed9d839a2d4fe6ff516e87fa8d3e656a0f99fa1f0105f73b3b9d19f |
| Cofactor | 0x1 |
| $j$-invariant | 0x6ebb653b932f9624f3cb17bda956ae317e2d850ae2d7662ead52e87ba272a41f060ce3279d0215c60fa2ab203be316c8e63508856621e690c00017fe748859c2 |
| Trace $t$ | 0x724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462c81 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff8dbefa3f5ed9d839a2d4fe6ff516e87fa8d3e656a0f99fa1f0105f73b3b9d19f |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x8000000000000000000000000000000000000000000000000000000000000000724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462aa1 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1e32fdf9a37b78e80a79f00a21d9b0e69cb4c2d06cd7974fcdac751309826c95dd42eafa496554fbdc2420d8d2d55040687840b1fe148bbf3571f9aac0f64c2485 |
| 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 | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x3', '0x3', '0x97b425ed097b425ed097b425ed097b425ed097b425ed097b425ed097b425ed08f3d8dcbce23197f875225cf677cf4c715fcbbba90aa31c024bed7a9c1760f87'] |
| (-)largest_factor_bitlen | 0x1fc |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2', '0x2', '0x5', '0x151', '0x37f7', '0x3f00dd41', '0x10f16d152b5c2e4ddebe085630bb00c81d6b0121acd68638a719eda493819d14ee6004791dfb29033bc82066162d2e8403686e89f4dcd024e3d'] |
| (-)largest_factor_bitlen | 0x1c9 |
| $\text{kn_factorization}(k=4)$ | |
| (+)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 | ['0x2', '0x2', '0x25', '0x45306eb3e45306eb3e45306eb3e45306eb3e45306eb3e45306eb3e45306eb3e41544a2f1d267ea8004fd829d61e2de8a31bea60550183a9cb91d9b6122e0fbb'] |
| (+)largest_factor_bitlen | 0x1fb |
| (-)factorization | ['0x2', '0x3', '0x2ab', '0x7335f1', '0x58d655f7de146df2bdbd1b913e4973ede116962df253fecd3194018be5e27e318f61760cb883373c943171dd39ad8fa561eb4a950a044c9490e4c3dd'] |
| (-)largest_factor_bitlen | 0x1df |
| $\text{kn_factorization}(k=6)$ | |
| (+)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 | ['0x2', '0x3', '0x3', '0x5', '0x5', '0x5', '0x60d', '0x2955b', '0x118567', '0x255205', '0x2ffd5136461', '0xd9ec86bbc4bb586cabd8d4f9136ecee372318407cb27f1738d58e85e7b3a3f32310f0e176e89e7b8d395ba5f20c406bf5d'] |
| (+)largest_factor_bitlen | 0x188 |
| (-)factorization | ['0x2', '0x2', '0x2', '0x61f1', '0x15bb9', '0x10d03b', '0x17dbe7', '0x2d78d3', '0x30654f40d0e2fabf6df86df2012db798a4085a3b781964d334036eea7f06074e5a5739331b609f96159d45e1bc82406e00c0ae575'] |
| (-)largest_factor_bitlen | 0x1a2 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x22d', '0x3805', '0x2705c7b', '0x8debcfc3', '0xbe08437dafdabf', '0x2771e87611b3fd71', '0x11812ebc276edd419', '0x31a7e6ed82b11e5902774945e991db31ecfe0b0a59593a77a9359ae3bd75315f'] |
| (+)largest_factor_bitlen | 0xfe |
| (-)factorization | ['0x3', '0x5', '0x7', '0xb', '0x35', '0x4484807cb931e3091ccf4696714e7dda40e1174a19bc64e2d8efa4eee43a3adb34d652a8e41f61ceb34abfe93a2a36c65eae0acfce00ed7943da7a65c9f29'] |
| (-)largest_factor_bitlen | 0x1f3 |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x3 |
| full | 0x7 |
| relative | 0x2 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0xa |
| full | 0xa |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462c81 |
| factorization | ['0x3', '0xd', '0x36bb', '0x8174f33', '0x1b190364c4ccf66fb29fbffbaafb4439ed78ef9d641d2e725d557'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x4d02065c848717f5860ff5de264f19634b3d2f932868b032538aecf67d936a22bd1505b69aab0423dbdf272d2aafbf9787bf4e01eb7440ca8e06553f09b7651e |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x5b7ef8f41d333719be0dbc124cb794eeec34eac5c4d8a4fa8ba62fbd5b70917cb107a2134ac0a111ad7db98dc14fdce93da013217ca6c3aa8f300dde65f4c4d5ce85db566f990917bbec44d20f64f16bfd583816af69537cd53a4966a9257dbd |
| 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 | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0xc', '0x1']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 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 | 0x1 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x2 |
| 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 | 0x2 |
| 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 | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x7 |
| relative | 0x7 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0x5 |
| relative | 0x5 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x8 |
| relative | 0x8 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462c81 |
| trace_factorization | ['0x3', '0xd', '0x36bb', '0x8174f33', '0x1b190364c4ccf66fb29fbffbaafb4439ed78ef9d641d2e725d557'] |
| number_of_factors | 0x5 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462c81 |
| trace_factorization | ['0x9203acf43142e19', '0x1676e3a47c169529d2799106a50fff36c838c943c8fd940e26c3711510ad85b53ac7836e06b05c4985f38f1c67aea2edde6185c3e7edf87ac5'] |
| number_of_factors | 0x2 |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0xde |
| expected | 0xff |
| ratio | 1.14865 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0xfec9 |
| expected | 0xfe80 |
| ratio | 0.99888 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xa8b3f1 |
| expected | 0xa8ac7f |
| ratio | 0.99983 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x724105c0a12627c65d2b01900ae91780572c19a95f06605e0fefa08c4c462e61 |
| ratio | 1.2972314489995918e+77 |
| distance 32 | 0x1 |
| distance 64 | 0x1f |
| $\text{multiples_x}(k=1)$ | |
| 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 | 0x7cfb50617f90081294fe87ba6f3d6c744a0185f7ff9413c5e77e164b06c30d137371b751278bac07a0b0eeff5bd6a9c298e09351df6e8e34688c6c66942f2175 |
| $\text{brainpool_overlap}()$ | |
| o | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1c |
| $\text{weierstrass}()$ | |
| a | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe1c |
| b | 0x879da |