Curve detail

Definition

Name secp128r2
Category secg
Description A randomly generated curve. [SEC2v1](https://www.secg.org/SEC2-Ver-1.0.pdf) states 'E was chosen verifiably at random as specified in ANSI X9.62 [1] from the seed'.
Field Prime (0xfffffffdffffffffffffffffffffffff)
Field bits 128
Form Weierstrass $y^2 = x^3 + ax + b$
Param $a$ 0xd6031998d1b3bbfebf59cc9bbff9aee1
Param $b$ 0x5eeefca380d02919dc2c6558bb6d8a5d
Generator $x$ 0x7b6aa5d85e572983e6fb32a7cdebc140
Generator $y$ 0x27b6916a894d3aee7106fe805fc34b44
Simulation seed 0x4d696e67687561517512d8f03431fce63b88f4

Characteristics

Order 0x3fffffff7fffffffbe0024720613b5a3
Cofactor 0x4
$j$-invariant 0x92a06b59be064918b6fdd808c5363c53
Trace $t$ 0x107ff6e37e7b12974
Embedding degree $k$ 0x2e8ba2e85d1745d1445d31c74a3ce13
CM discriminant -0xbbf04b2917c794a45cddd32f93b868db

Traits

$\text{cofactor}()$
order 0x3fffffff7fffffffbe0024720613b5a3
cofactor 0x4
$\text{discriminant}()$
cm_disc -0xbbf04b2917c794a45cddd32f93b868db
factorization ['0x2', '0x2', '0x17', '0x2f', '0x2c81d87d4c70a03da895275b131db3']
max_conductor 0x2
$\text{twist_order}(deg=1)$
twist_cardinality 0xfffffffe0000000107ff6e37e7b12974
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0xfffffffc00000003fffffffffffffffd103ed35ba0e1ad6e8c88b341b11e5c94
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x3', '0x65', '0x97', '0x26b', '0x425', '0x96b', '0x3e29e92cd491936c88319']
(+)largest_factor_bitlen 0x52
(-)factorization ['0x5', '0x7', '0x7', '0x119', '0x1c91e07', '0x3123826f3', '0x2c702b182c2c4b']
(-)largest_factor_bitlen 0x36
$\text{kn_factorization}(k=2)$
(+)factorization ['0xd', '0x1541f6b', '0x6dda4f1f', '0x4514c98d4eae3f4529']
(+)largest_factor_bitlen 0x47
(-)factorization ['0x3', '0x3', '0x3', '0x3', '0x11', '0x35', '0xce3', '0x1ce65b', '0x13c0a5138a91e7b1b40eb']
(-)largest_factor_bitlen 0x51
$\text{kn_factorization}(k=3)$
(+)factorization ['0x25', '0x59', '0x19c42f', '0x7e72716a5', '0x4b0f554dae63e09b']
(+)largest_factor_bitlen 0x3f
(-)factorization ['0xb', '0x17', '0x1d', '0x3b23', '0x740081dbeeb5777416a0eba919']
(-)largest_factor_bitlen 0x67
$\text{kn_factorization}(k=4)$
(+)factorization ['0x3', '0x5', '0x1f', '0x71f8388d1bafd23', '0x4f24dbc71659fcab']
(+)largest_factor_bitlen 0x3f
(-)factorization ['0x13', '0x977', '0x1b717b', '0x351e0de82d8112311be3d9e9']
(-)largest_factor_bitlen 0x5e
$\text{kn_factorization}(k=5)$
(+)factorization ['0x147e57249003', '0x3e757257aac3a2012e403f']
(+)largest_factor_bitlen 0x56
(-)factorization ['0x3', '0x862b1', '0x507f27b072995', '0xa1d0d8ff0eb6e95']
(-)largest_factor_bitlen 0x3c
$\text{kn_factorization}(k=6)$
(+)factorization ['0x7', '0x151', '0x5cf7908c75b6793', '0x1cb0065f9b83dfe5']
(+)largest_factor_bitlen 0x3d
(-)factorization ['0x5', '0x133333330cccccccb9000aef01d2b67db']
(-)largest_factor_bitlen 0x81
$\text{kn_factorization}(k=7)$
(+)factorization ['0x3', '0x3', '0xfb', '0x107', '0x1618f87cd0b67b', '0x8f208bedc58d523']
(+)largest_factor_bitlen 0x3c
(-)factorization ['0x3b', '0x2b13', '0x9a61', '0x6de13', '0x2b966f4fa725f3860d81']
(-)largest_factor_bitlen 0x4e
$\text{kn_factorization}(k=8)$
(+)factorization ['0xb', '0xd3', '0xe1e38b13f6bc2acab2e77709e67651']
(+)largest_factor_bitlen 0x78
(-)factorization ['0x3', '0x7', '0x2e3', '0xb29', '0x6cc5ae3e3', '0x71fe31d476a450f493']
(-)largest_factor_bitlen 0x47
$\text{torsion_extension}(l=2)$
least 0x1
full 0x2
relative 0x2
$\text{torsion_extension}(l=3)$
least 0x4
full 0x4
relative 0x1
$\text{torsion_extension}(l=5)$
least 0x18
full 0x18
relative 0x1
$\text{torsion_extension}(l=7)$
least 0x6
full 0x6
relative 0x1
$\text{torsion_extension}(l=11)$
least 0x78
full 0x78
relative 0x1
$\text{torsion_extension}(l=13)$
least 0x3
full 0xc
relative 0x4
$\text{torsion_extension}(l=17)$
least 0x30
full 0x30
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0x107ff6e37e7b12974
factorization ['0x2', '0x2', '0x3', '0x15fff3d9fdf96e1f']
$\text{conductor}(deg=3)$
ratio_sqrt 0x103ed355a0e1ad6e8c88b341b11e5c91
factorization ['0x7', '0x11', '0x8354953', '0x50be5c23', '0xd7fd0473a47da0f']
$\text{conductor}(deg=4)$
ratio_sqrt 0xf73ead85d21d11c1856d248950edd591e960c5fa2278abd8
factorization ['0x2', '0x2', '0x2', '0x3', '0x13', '0x9d31', '0x15fff3d9fdf96e1f', '0xa467a09a4bad6308e0b6b9aa1dd']
$\text{embedding}()$
embedding_degree_complement 0x16
complement_bit_length 0x5
$\text{class_number}()$
upper 0x181d06a4c4cd3e6de9
lower 0x40
$\text{small_prime_order}(l=2)$
order None
complement_bit_length None
$\text{small_prime_order}(l=3)$
order 0x3fffffff7fffffffbe0024720613b5a2
complement_bit_length 0x2
$\text{small_prime_order}(l=5)$
order 0x1fffffffbfffffffdf0012390309dad1
complement_bit_length 0x3
$\text{small_prime_order}(l=7)$
order 0x3fffffff7fffffffbe0024720613b5a2
complement_bit_length 0x2
$\text{small_prime_order}(l=11)$
order 0x3fffffff7fffffffbe0024720613b5a2
complement_bit_length 0x2
$\text{small_prime_order}(l=13)$
order 0x3fffffff7fffffffbe0024720613b5a2
complement_bit_length 0x2
$\text{division_polynomials}(l=2)$
factorization [['0x1', '0x1'], ['0x2', '0x1']]
len 0x2
$\text{division_polynomials}(l=3)$
factorization [['0x2', '0x2']]
len 0x1
$\text{division_polynomials}(l=5)$
factorization [['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 0x2
depth 0x0
$\text{volcano}(l=11)$
crater_degree 0x0
depth 0x0
$\text{volcano}(l=13)$
crater_degree 0x2
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 0x6
full 0x6
relative 0x1
$\text{isogeny_extension}(l=7)$
least 0x1
full 0x3
relative 0x3
$\text{isogeny_extension}(l=11)$
least 0xc
full 0xc
relative 0x1
$\text{isogeny_extension}(l=13)$
least 0x1
full 0xc
relative 0xc
$\text{isogeny_extension}(l=17)$
least 0x3
full 0x3
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x4
full 0x4
relative 0x1
$\text{trace_factorization}(deg=1)$
trace 0x107ff6e37e7b12974
trace_factorization ['0x2', '0x2', '0x3', '0x15fff3d9fdf96e1f']
number_of_factors 0x3
$\text{trace_factorization}(deg=2)$
trace 0x107ff6e37e7b12974
trace_factorization ['0x2', '0x13', '0x9d31', '0xa467a09a4bad6308e0b6b9aa1dd']
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 0x41
expected 0x40
ratio 0.98462
$\text{hamming_x}(weight=2)$
x_coord_count 0xf74
expected 0xfe0
ratio 1.0273
$\text{hamming_x}(weight=3)$
x_coord_count 0x2992c
expected 0x29ac0
ratio 1.00237
$\text{square_4p1}()$
p 0x1
order 0x7
$\text{pow_distance}()$
distance 0x20000000107ff6e37e7b12974
ratio 2147483646.74219
distance 32 0xc
distance 64 0xc
$\text{multiples_x}(k=1)$
Hx 0x7b6aa5d85e572983e6fb32a7cdebc140
bits 0x7f
difference 0x1
ratio 1.00794
$\text{multiples_x}(k=2)$
Hx 0xf69ff511dbd13171f8d75488000fbca2
bits 0x80
difference 0x0
ratio 1.01587
$\text{multiples_x}(k=3)$
Hx 0x54321ece622a59689ae8a38844655ebd
bits 0x7f
difference 0x1
ratio 1.00794
$\text{multiples_x}(k=4)$
Hx 0x161ff7528b899b2d0c28607ca52c5b86
bits 0x7d
difference 0x3
ratio 0.99206
$\text{multiples_x}(k=5)$
Hx 0x7f8a99721379f16d21efa0004302e7a
bits 0x7b
difference 0x5
ratio 0.97619
$\text{multiples_x}(k=6)$
Hx 0x95a708d73ce03fd53ea8161b53e211af
bits 0x80
difference 0x0
ratio 1.01587
$\text{multiples_x}(k=7)$
Hx 0x15a8efa245d73c333474af00f24ececb
bits 0x7d
difference 0x3
ratio 0.99206
$\text{multiples_x}(k=8)$
Hx 0x4a622baa13e39176f9cad031b249a725
bits 0x7f
difference 0x1
ratio 1.00794
$\text{multiples_x}(k=9)$
Hx 0xd69d80396264fd174a92e872cc9e4ba0
bits 0x80
difference 0x0
ratio 1.01587
$\text{multiples_x}(k=10)$
Hx 0xf08568a9d55bdc725287f5da65b138c0
bits 0x80
difference 0x0
ratio 1.01587
$\text{x962_invariant}()$
r 0x1def19fd392155bb3f430f1ad91327d8
$\text{brainpool_overlap}()$
o None
$\text{weierstrass}()$
a 0xd6031998d1b3bbfebf59cc9bbff9aee1
b 0x5eeefca380d02919dc2c6558bb6d8a5d