Curve detail

Definition

Name B-163 (nist/B-163, secg/sect163r2, x962/ansit163r2)
Category nist
Field Binary
Field polynomial $x^{163} + x^{7} + x^{6} + x^{3} + 1$
Field bits 163
Form Weierstrass $y^2 = x^3 + ax + b$
Param $a$ 0x000000000000000000000000000000000000000001
Param $b$ 0x020a601907b8c953ca1481eb10512f78744a3205fd
Generator $x$ 0x03f0eba16286a2d57ea0991168d4994637e8343e36
Generator $y$ 0x00d51fbc6c71a0094fa2cdd545b11c5c0c797324f1
Simulation seed 0x85e25bfe5c86226cdb12016f7553f9d0e693a268

Characteristics

Order 0x40000000000000000000292fe77e70c12a4234c33
Cofactor 0x2
Trace $t$ -0x525fcefce182548469865

Traits

$\text{cofactor}()$
order 0x40000000000000000000292fe77e70c12a4234c33
cofactor 0x2
$\text{discriminant}()$
cm_disc -0x57e7b8ab87d2e65f2ddc9a202f211399d7e0be827
factorization ['0x599', '0x1d5447', '0xd581ad22b084fcf', '0xa45a43146d75947be7']
max_conductor 0x1
$\text{twist_order}(deg=1)$
twist_cardinality 0x7fffffffffffffffffffada031031e7dab7b9679c
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0x40000000000000000000000000000000000000000a8184754782d19a0d22365dfd0deec66281f417da
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x5', '0x5', '0x53111d5f4d', '0x14005e422fc0f', '0xc9f53a968ea6840455']
(+)largest_factor_bitlen 0x48
(-)factorization ['0x3', '0x3', '0x7', '0x1eebd', '0x7a22a6381', '0x2341f07239a8313a92bf6425e77']
(-)largest_factor_bitlen 0x6a
$\text{kn_factorization}(k=2)$
(+)factorization ['0x3', '0x25', '0x49d', '0xd345e881', '0x8be8abf699', '0x11bc046f5b48629b7777']
(+)largest_factor_bitlen 0x4d
(-)factorization ['0xd', '0x15d', '0xe71dd94ae03b95b2054e6418b4fd3eee2528a3']
(-)largest_factor_bitlen 0x98
$\text{kn_factorization}(k=3)$
(+)factorization ['0x18d2e69', '0x119c2ddd', '0xe0e00072c0de4319e05f5a8ecbf7']
(+)largest_factor_bitlen 0x70
(-)factorization ['0x13', '0x1435e50d79435e50d79442e6b4e48f656ba91e22b']
(-)largest_factor_bitlen 0xa1
$\text{kn_factorization}(k=4)$
(+)factorization ['0x4f', '0x1dcc1185', '0x7489862f5', '0x8a693955fe67', '0xe23b3c00e2c9']
(+)largest_factor_bitlen 0x30
(-)factorization ['0x3', '0x5', '0xb', '0xa88f', '0x4ef55171af5d51', '0xf47a05ae7e3e76264cacb5']
(-)largest_factor_bitlen 0x58
$\text{kn_factorization}(k=5)$
(+)factorization ['0x3', '0x16db', '0x469d', '0x21d6e0640dff58a7e5acca1e6f8263d271b']
(+)largest_factor_bitlen 0x8a
(-)factorization ['0x9c2b', '0x5c84bd', '0x4ea9dc15', '0x24e742f3ad246392306f7f1f7']
(-)largest_factor_bitlen 0x62
$\text{kn_factorization}(k=6)$
(+)factorization ['0x5', '0x7', '0x11', '0x17', '0x175', '0x5d916a569904ae0b', '0x1afa3685b5f6c8d68b707']
(+)largest_factor_bitlen 0x51
(-)factorization ['0x498aea9889527', '0x17ac0777146141', '0x70ef452f5749a825']
(-)largest_factor_bitlen 0x3f
$\text{kn_factorization}(k=7)$
(+)factorization ['0xb', '0x97', '0x14a2b3', '0x2567660d7fa6fb775', '0x2dcd571e705d13d469']
(+)largest_factor_bitlen 0x46
(-)factorization ['0x3', '0x59f', '0x1c7f172e2c63', '0x47ed7c609b0f', '0x6a2dda8d8a53ea51']
(-)largest_factor_bitlen 0x3f
$\text{kn_factorization}(k=8)$
(+)factorization ['0x3', '0x3', '0x71c71c71c71c71c71c72105529c456acbce77a3e9']
(+)largest_factor_bitlen 0xa3
(-)factorization ['0x7', '0x1d', '0xb3cfcd713', '0x72e826220654ed2b92c56b6e6229b3f']
(-)largest_factor_bitlen 0x7b
$\text{torsion_extension}(l=2)$
least None
full None
relative None
$\text{torsion_extension}(l=3)$
least 0x8
full 0x8
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 0xa
relative 0x2
$\text{torsion_extension}(l=13)$
least 0xa8
full 0xa8
relative 0x1
$\text{torsion_extension}(l=17)$
least 0x30
full 0x30
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0x525fcefce182548469865
factorization ['0x5', '0x1d', '0x17141', '0x181a9', '0x78f49', '0x8da765']
$\text{conductor}(deg=3)$
ratio_sqrt 0x128184754782d19a0d22365dfd0deec66281f417d9
factorization ['0x11', '0x17', '0x32989', '0x3d4e2aad0fbb92c8ca6421316dff18971e7']
$\text{conductor}(deg=4)$
ratio_sqrt 0x3616afc51c362550fcbd45148799e7e752b88b52087fe8aa9cbbd6c0cc409d
factorization ['0x3', '0x3', '0x5', '0x7', '0x1d', '0x2b', '0x3a9', '0x17141', '0x181a9', '0x78f49', '0x8da765', '0x44061fffb', '0x1580162e81', '0xc279032db4af7be1f7']
$\text{embedding}()$
embedding_degree_complement 0x1
complement_bit_length 0x1
$\text{class_number}()$
upper 0x54042c4b1aae11f5f95a40
lower 0x5ff8
$\text{small_prime_order}(l=2)$
order None
complement_bit_length None
$\text{small_prime_order}(l=3)$
order 0x200000000000000000001497f3bf386095211a619
complement_bit_length 0x2
$\text{small_prime_order}(l=5)$
order 0x40000000000000000000292fe77e70c12a4234c32
complement_bit_length 0x1
$\text{small_prime_order}(l=7)$
order 0x40000000000000000000292fe77e70c12a4234c32
complement_bit_length 0x1
$\text{small_prime_order}(l=11)$
order 0x200000000000000000001497f3bf386095211a619
complement_bit_length 0x2
$\text{small_prime_order}(l=13)$
order 0x200000000000000000001497f3bf386095211a619
complement_bit_length 0x2
$\text{division_polynomials}(l=2)$
factorization [['0x1', '0x2']]
len 0x1
$\text{division_polynomials}(l=3)$
factorization [['0x4', '0x1']]
len 0x1
$\text{division_polynomials}(l=5)$
factorization [['0x4', '0x3']]
len 0x1
$\text{volcano}(l=2)$
crater_degree 0x2
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 0x0
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 None
full None
relative None
$\text{isogeny_extension}(l=3)$
least 0x4
full 0x4
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 0xa
relative 0xa
$\text{isogeny_extension}(l=13)$
least 0xe
full 0xe
relative 0x1
$\text{isogeny_extension}(l=17)$
least 0x3
full 0x3
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x1
full 0x12
relative 0x12
$\text{trace_factorization}(deg=1)$
trace -0x525fcefce182548469865
trace_factorization ['0x5', '0x1d', '0x17141', '0x181a9', '0x78f49', '0x8da765']
number_of_factors 0x6
$\text{trace_factorization}(deg=2)$
trace -0x525fcefce182548469865
trace_factorization ['0x3', '0x3', '0x7', '0x2b', '0x3a9', '0x44061fffb', '0x1580162e81', '0xc279032db4af7be1f7']
number_of_factors 0x7
$\text{isogeny_neighbors}(l=2)$
len 0x3
$\text{isogeny_neighbors}(l=3)$
len 0x0
$\text{isogeny_neighbors}(l=5)$
len 0x0
$\text{q_torsion}()$
Q_torsion 0x1
$\text{hamming_x}(weight=1)$
x_coord_count 0xa3
expected 0x52
ratio 0.50307
$\text{hamming_x}(weight=2)$
x_coord_count 0x3393
expected 0x1a1b
ratio 0.50617
$\text{hamming_x}(weight=3)$
x_coord_count 0xacfd1
expected 0x581b2
ratio 0.50932
$\text{square_4p1}()$
p 0x1
order 0x1
$\text{pow_distance}()$
distance 0x525fcefce182548469866
ratio 1.8785300127524536e+24
distance 32 0x6
distance 64 0x1a
$\text{multiples_x}(k=1)$
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 0x0
$\text{brainpool_overlap}()$
o None
$\text{weierstrass}()$
a None
b None