Curve detail

Definition

Name w-254-mont
Category nums
Description Original nums curve from https://eprint.iacr.org/2014/130.pdf
Field Prime (0x3f80ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff)
Field bits 254
Form Weierstrass $y^2 = x^3 + ax + b$
Param $a$ 0x3f80fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc
Param $b$ -0x2f72

Characteristics

Order 0x3f80ffffffffffffffffffffffffffffeb818bea0da375c06fa419c4af8df83f
Cofactor 0x1
$j$-invariant 0x241bdb147c4142e61d7b9951a5890233884e3b890c9b6933f01b832cb776362
Trace $t$ 0x147e7415f25c8a3f905be63b507207c1
Embedding degree $k$ 0xcb36666666666666666666666666666624d1bfb9c53e459afed9ec0efe931a6
CM discriminant -0xfc5fff6617e311c874f4297b4c0b5f269ab3a1cac911feec4904296677dfe07b

Traits

$\text{cofactor}()$
order 0x3f80ffffffffffffffffffffffffffffeb818bea0da375c06fa419c4af8df83f
cofactor 0x1
$\text{discriminant}()$
cm_disc -0xfc5fff6617e311c874f4297b4c0b5f269ab3a1cac911feec4904296677dfe07b
factorization ['0x7', '0x3b', '0xbbff7', '0x56ed308d', '0x2735a0963c47f995916ae583e16403e6c3dfb4c9093a59a345']
max_conductor 0x1
$\text{twist_order}(deg=1)$
twist_cardinality 0x3f810000000000000000000000000000147e7415f25c8a3f905be63b507207c1
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0xfc0bf00ffffffffffffffffffffffffffffffffffffffffffffffffffffffff03a00099e81cee378b0bd684b3f4a0d9654c5e3536ee0113b6fbd69988201f85
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x2', '0x2', '0x2', '0x2', '0x2', '0x2', '0xcd857bb', '0xacd9b459', '0x29ab55d7718e45', '0x1cd9a8b1a7707ded', '0x63ca4594a554a3589ab']
(+)largest_factor_bitlen 0x4b
(-)factorization ['0x2', '0x3', '0x5', '0xb', '0x732b27f', '0x6d812e8142ac0506f0754dbd83650dca2565b6ae64964eb4416c60d']
(-)largest_factor_bitlen 0xdb
$\text{kn_factorization}(k=2)$
(+)factorization ['0x3', '0x3', '0x29', '0x63d9', '0xac27', '0x4322e1e9', '0x8eb1df657b', '0x43d99a0980e9', '0x65a8472eb8d9', '0x554c4cdff21d1b']
(+)largest_factor_bitlen 0x37
(-)factorization ['0x7', '0xd', '0x1d', '0x15b', '0x557', '0x23fb6f', '0x8ebbddff9', '0x15b88f8ca0a498b5643776d6772374e1514b8fb2959']
(-)largest_factor_bitlen 0xa9
$\text{kn_factorization}(k=3)$
(+)factorization ['0x2', '0x49d50ad1f8bffe1', '0x14a48618ca8cd7bd814766833f2f43e5c228cbacdab1f07c3f']
(+)largest_factor_bitlen 0xc5
(-)factorization ['0x2', '0x2', '0x2f', '0x125', '0x2e3', '0x4e84b8ba4f86c3523629ad42c6a89d2376c17fa917f7e8190ffbbd132f']
(-)largest_factor_bitlen 0xe7
$\text{kn_factorization}(k=4)$
(+)factorization ['0x5', '0x4006f9', '0xbca2125', '0x5278c0fd', '0x9a05300de45029', '0x58e477a0b85d123e9a010a811247a9']
(+)largest_factor_bitlen 0x77
(-)factorization ['0x3', '0x74b', '0x9ba21a8dd', '0x430dcb422bf', '0x6ea49f00c26251', '0xa8b221f82592bce37920d5410259']
(-)largest_factor_bitlen 0x70
$\text{kn_factorization}(k=5)$
(+)factorization ['0x2', '0x2', '0x3', '0x7', '0x25cf2069268f', '0x4bbb8cb173711', '0x5683ddc60f7365bc13741e394d6e126ba6fecad']
(+)largest_factor_bitlen 0x9b
(-)factorization ['0x2', '0x13', '0xca375', '0xa9403ef43b2e4f8bd3d683c5bb20c1fb1850174c23944878f10912a573']
(-)largest_factor_bitlen 0xe8
$\text{kn_factorization}(k=6)$
(+)factorization ['0x1d661d6c241c75', '0xcf5e17a1ca4519b2da594c10edee6fa41726a516dab8f75382f']
(+)largest_factor_bitlen 0xcc
(-)factorization ['0x5', '0x11', '0x18b9', '0xe595f1a31019229', '0x33c1e6e1128a3fe713f6d7170db4f5d0840df943bf475']
(-)largest_factor_bitlen 0xb2
$\text{kn_factorization}(k=7)$
(+)factorization ['0x2', '0x43', '0x49', '0x17b', '0x283', '0x6b59', '0x1d80c81', '0x9414067', '0x58332fc1', '0xd04303ed3', '0x136732b89e8b', '0x14946ae87e629']
(+)largest_factor_bitlen 0x31
(-)factorization ['0x2', '0x2', '0x2', '0x3', '0x3', '0x3', '0x2e1d342991a317', '0x15b72abfe1f4715', '0x86af87e8cd193bd34d1f4992dd318df1ea7']
(-)largest_factor_bitlen 0x8c
$\text{kn_factorization}(k=8)$
(+)factorization ['0x3', '0x89', '0x1ec07', '0x338e6de3', '0x33184712edb84faab4fcc03073564062fadf9f99993e8fb8c6f']
(+)largest_factor_bitlen 0xca
(-)factorization ['0xe5', '0x943', '0x6fed1', '0x30b3841733', '0x406700e925f9dd26c62a7', '0xb728efc8faaa136fdfa136895']
(-)largest_factor_bitlen 0x64
$\text{torsion_extension}(l=2)$
least 0x3
full 0x3
relative 0x1
$\text{torsion_extension}(l=3)$
least 0x8
full 0x8
relative 0x1
$\text{torsion_extension}(l=5)$
least 0x6
full 0x6
relative 0x1
$\text{torsion_extension}(l=7)$
least 0x6
full 0x7
relative 0x1
$\text{torsion_extension}(l=11)$
least 0x5
full 0x5
relative 0x1
$\text{torsion_extension}(l=13)$
least 0x18
full 0x18
relative 0x1
$\text{torsion_extension}(l=17)$
least 0x18
full 0x18
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0x147e7415f25c8a3f905be63b507207c1
factorization ['0xd', '0x49', '0x513b24d790687', '0x116c3bbaf7caeebfb']
$\text{conductor}(deg=3)$
ratio_sqrt 0x3ddcff6617e311c874f4297b4c0b5f269ab3a1cac911feec4904296677dfe07e
factorization ['0x2', '0x5', '0x11', '0xb3', '0x853b92f46801e642c588621635c4de382fae2d03cb743465a2739a5c7031']
$\text{conductor}(deg=4)$
ratio_sqrt 0xa094511053de2435b41ee1728c2e21d69e845624a67313a74175a131d577da77c5336a3629d3cdb7f14662f8695a93d
factorization ['0x3', '0xd', '0x49', '0x102b', '0x3d2b7ed78f84b', '0x513b24d790687', '0x116c3bbaf7caeebfb', '0xad12af1dea2f4d66f2bc104f57f9cc1ce1f2a3b684bfe1d7']
$\text{embedding}()$
embedding_degree_complement 0x5
complement_bit_length 0x3
$\text{class_number}()$
upper 0x3813ae3ecb9accb14bc3cc8bc4b1347a43
lower 0x4d276b
$\text{small_prime_order}(l=2)$
order 0x659b33333333333333333333333333331268dfdce29f22cd7f6cf6077f498d3
complement_bit_length 0x4
$\text{small_prime_order}(l=3)$
order 0x3f80ffffffffffffffffffffffffffffeb818bea0da375c06fa419c4af8df83e
complement_bit_length 0x1
$\text{small_prime_order}(l=5)$
order 0x659b33333333333333333333333333331268dfdce29f22cd7f6cf6077f498d3
complement_bit_length 0x4
$\text{small_prime_order}(l=7)$
order 0x152afffffffffffffffffffffffffffff92b2ea359e127402536b3418fd9fd6a
complement_bit_length 0x2
$\text{small_prime_order}(l=11)$
order 0xcb36666666666666666666666666666624d1bfb9c53e459afed9ec0efe931a6
complement_bit_length 0x3
$\text{small_prime_order}(l=13)$
order 0x5c5e8ba2e8ba2e8ba2e8ba2e8ba2e8ba10bc6e6bb6bf36e95c9199d8452b97a
complement_bit_length 0x4
$\text{division_polynomials}(l=2)$
factorization [['0x3', '0x1']]
len 0x1
$\text{division_polynomials}(l=3)$
factorization [['0x4', '0x1']]
len 0x1
$\text{division_polynomials}(l=5)$
factorization [['0x3', '0x4']]
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 0x0
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 0x4
full 0x4
relative 0x1
$\text{isogeny_extension}(l=5)$
least 0x3
full 0x3
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 0x3
full 0x3
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x1
full 0x9
relative 0x9
$\text{trace_factorization}(deg=1)$
trace 0x147e7415f25c8a3f905be63b507207c1
trace_factorization ['0xd', '0x49', '0x513b24d790687', '0x116c3bbaf7caeebfb']
number_of_factors 0x4
$\text{trace_factorization}(deg=2)$
trace 0x147e7415f25c8a3f905be63b507207c1
trace_factorization ['0x3', '0x102b', '0x3d2b7ed78f84b', '0xad12af1dea2f4d66f2bc104f57f9cc1ce1f2a3b684bfe1d7']
number_of_factors 0x4
$\text{isogeny_neighbors}(l=2)$
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 0x81
expected 0x7f
ratio 0.9845
$\text{hamming_x}(weight=2)$
x_coord_count 0x3fa4
expected 0x3ec1
ratio 0.98607
$\text{hamming_x}(weight=3)$
x_coord_count 0x149cca
expected 0x14977e
ratio 0.999
$\text{square_4p1}()$
p 0x1
order 0x1
$\text{pow_distance}()$
distance 0x7f0000000000000000000000000000147e7415f25c8a3f905be63b507207c1
ratio 128.00787
distance 32 0x1
distance 64 0x1
$\text{multiples_x}(k=1)$
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 0x257b46b4007528497a9d0fc152d5bed262c2a58eb7f26dd99a8e09ff2b9a21
$\text{brainpool_overlap}()$
o 0x1ffffffffffffffffffffffd
$\text{weierstrass}()$
a 0x3f80fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc
b -0x2f72