ECB - Elliptic Curve Builder - is a generator of ordinary elliptic curves.
The curves over GF(P), GF(2N) and GF(3N) are built using the
so-called complex multiplication method. Even if, for some reasons, one does
not trust the curves produced with ECB, they remain useful in order
to test and/or to tune ECC applications.
Executable for Linux 64-bit (Ubuntu 18.04/x86-64 architecture)
Compiled with Free Pascal 3.0.4 and
Lazarus 1.8.2
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Properties of a curve created with ECB
- over GF(P)
- equation y2 = x3 + Ax + B;
- the order is U = R*K with R prime and K < R;
- the binary size of the prime modulus P may be any in 30..1536.
- over GF(2N)
- equation y2 + xy = x3 + Ax2 + B;
- the order is U = R*K with R prime and K < R;
- the field degree N may be any in 30..1024;
- the basis of the field GF(2N) may be polynomial or normal.
- over GF(3N)
- equation y2 = x3 + Ax2 + B;
- the order is U = R*K with R prime and K < R;
- the field degree N may be any in 20..768;
- the basis of the field GF(3N) may be polynomial or normal.
Over GF(P)
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Parameters
----------
P = 81598516213282754316057565591253440513595901172568168425578611827
Discriminant = -261762
Class number = 288
Order U = R*K with R BPSW-prime
-------------------------------
U = 81598516213282754316057565591253746600275322583683770317215734358
R = 40799258106641377158028782795626873300137661291841885158607867179
K = 2
U binary size = 216
R binary size = 215
K binary size = 2
MOV condition
-------------
(P**e mod R) <> 1 for all e in 1..200
Field GF(P)
-----------
P = 81598516213282754316057565591253440513595901172568168425578611827
J-invariant
-----------
J = 39478255298952272494102096060286924102007387693101885025049078757
Curve (y**2 = x**3 + Ax + B) of order R*K
-----------------------------------------
R = 40799258106641377158028782795626873300137661291841885158607867179
K = 2
A = -3
B = 39203769498218508684097448848632530914621544507728641390082973841
Base point G (of order R)
-------------------------
X = 43476061527435728668477113342291061267119890801047274484944180244
Y = 43201194101978834386880648267297883678441846149320875637805690899
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Over GF(2N)
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Parameters
----------
Field degree = 223
Discriminant = -679351
Class number = 446
Order U = R*K with R BPSW-prime
-------------------------------
U = 13479973333575319897333507543509814922446042298823059087804060576348
R = 3369993333393829974333376885877453730611510574705764771951015144087
K = 4
U binary size = 223
R binary size = 221
K binary size = 3
K factorization = 2**2
MOV condition
-------------
(2**e mod R) <> 1 for all e in 1..2230
Field GF(2**223)
----------------
Field polynomial = [223,33,0]
Basis type = Polynomial
J-invariant
-----------
J = 16#2F7C2925C50877B0BF396F8A767F2E855CB87DAA112FFEE8B654CA46#
Curve (y**2 + xy = x**3 + Ax**2 + B) of order R*K
-------------------------------------------------
R = 3369993333393829974333376885877453730611510574705764771951015144087
K = 4
A = 16#0#
B = 16#6DFD3C1C5CB789DA8ABB326FA25309D5D06DE2A08C77D524AE690461#
Base point G (of order R)
-------------------------
X = 16#2700F05CE7FCA11E0CE5D4188314A6100447744E68FC7E7D523BDF42#
Y = 16#164C2CF26E46F18C0D5ABACC53DB4216690DEE801AEDAC2C80ABEB8F#
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Over GF(3N)
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Parameters
----------
Field degree = 127
Discriminant = -427067
Class number = 127
Order U = R*K with R BPSW-prime
-------------------------------
U = 3930061525912861057173624287137094778397646624162832523320429
R = 1310020508637620352391208095712364926132548874720944174440143
K = 3
U binary size = 202
R binary size = 200
K binary size = 2
MOV condition
-------------
(3**e mod R) <> 1 for all e in 1..1143
Field GF(3**127)
----------------
Field polynomial = [1,127,2,126,2,74,2,0]
Basis type = Normal
Field multiplicative identity
-----------------------------
I = 9#1444444444444444444444444444444444444444444444444444444444444444#
J-invariant
-----------
J = 9#2546211278166744651571743412152324532012305117470187544380405724#
Curve (y**2 = x**3 + Ax**2 + B) of order R*K
--------------------------------------------
R = 1310020508637620352391208095712364926132548874720944174440143
K = 3
A = 9#1444444444444444444444444444444444444444444444444444444444444444#
B = 9#2503707542263565644442537604567155105413720456058768265246780510#
Base point G (of order R)
-------------------------
X = 9#1802145614788372537170244845128202454170083174688378761362676814#
Y = 9#0027466558584743801174531346588086778106014481028255255865764138#
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v2.0.5 (April 28, 2019)
Previous changes
The ECB software may be used free of charge but it might be a good idea to read the
End-User License Agreement before downloading and using it.
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