ECB - Elliptic Curve Builder

Overview
Examples of use
Changes

Overview

ECB is a generator of elliptic curves that are intended for cryptographical purposes. The curves, over GF(P) or over GF(2ⁿ), are generated using the so-called complex multiplication method.

Executable for use under Win XP on 32-bit processors

Properties of a curve created with ECB

Over GF(P):
- equation y² = x³ + Ax + B;
- the order is U = R*S with S smooth and R a BPSW (Baillie, Pomerance, Selfridge & Wagstaff) pseudoprime;
- the binary size of the prime modulus P may be any in 33..512.
Over GF(2ⁿ):
- equation y² + xy = x³ + Ax² + B;
- the order is U = R*S with S smooth and R a BPSW pseudoprime;
- the field degree n may be any in 33..512;
- the basis of the Galois field GF(2ⁿ) may be standard or normal (ECB works with any normal basis, i.e., not with Gaussian normal ones only).

Examples of use

Over GF(P)

Parameters
----------
P = 1019822425152210877949330633665510840063159640183534239657159
Discriminant = -4795
ClassNumber = 12
GenusNumber = 4

Order U = R*S with R pseudoprime
--------------------------------
U = 1019822425152210877949330633665852523887456813016356781051489
R = 1019822425152210877949330633665852523887456813016356781051489
S = 1
U binary size = 200
R binary size = 200
S binary size = 1

MOV condition
-------------
Critical threshold = 9
Checked up to 90

Class polynomial factors (mod P)
--------------------------------
4 degree-3 factors
Q[0,3] = 1
Q[0,2] = -51301909889103105518557739260410974945123278680757771165551
Q[0,1] = 203140187800760456277975346098939541777078420050677780522614
Q[0,0] = 157393450616952842640789376196480017027311559331146838010293
Q[1,3] = 1
Q[1,2] = -221464647713733972198313197735350330768209631792833519826722
Q[1,1] = -493391488304099658627321820776239905549208696629748816608785
Q[1,0] = 176474156382251622839199572981140811737773836081256699007592
Q[2,3] = 1
Q[2,2] = 417737094469226266125825449978473286079956234071397612902226
Q[2,1] = -478901429125546897311130756345914755871078558287527942999170
Q[2,0] = -431729822970909185721018313233740050908158871225181869919599
Q[3,3] = 1
Q[3,2] = -144970536866389188408954512982711980366623323566416508887875
Q[3,1] = -250669695523324778288853402642295720419950805763842694166244
Q[3,0] = 97862215971704720241029364056119222143073482409592172670972

Root of Q over GF(P)
--------------------
Z = 190987801642575392101911874518786930511677216775326746406031

Curve of order R*S over GF(P)
-----------------------------
P = 1019822425152210877949330633665510840063159640183534239657159
R = 1019822425152210877949330633665852523887456813016356781051489
S = 1
A = -3
B = 506780569080129316469714699295613600568636592675898205448554

Point of Order R
----------------
X = 688276248190153899136892823075396429936228002705782769474491
Y = 315213794215586947062208847855771128161518503723629003807235

Over GF(2ⁿ)

Parameters
----------
Field degree = 191
Discriminant = -564127

Order U = R*S with R pseudoprime
--------------------------------
U = 3138550867693340381917894711531278116929969738635840702322
R = 1569275433846670190958947355765639058464984869317920351161
S = 2
U binary size = 191
R binary size = 190
S binary size = 2

MOV condition
-------------
Critical threshold = 9
Checked up to 90

Class polynomial (mod 2)
------------------------
Degree = 191
W = EC92F4A31A57C50F6031A2FA59ED0350A61C7BB051D1EBAD
W is irreducible over GF(2)

Field GF(2^191)
---------------
Field polynomial = [191,9,0]
Basis type = Standard

Root of W over GF(2^191)
------------------------
Z = E3E9AF02AF5BB0F5496B7DA70744FFDBB39A1007E52BC13

Curve of order R*S over GF(2^191)
---------------------------------
R = 1569275433846670190958947355765639058464984869317920351161
S = 2
A = 1
B = 2AC03DC6A89D81EBC78E7A190E7853DE38E9DEEEAC364006

Point of order R
----------------
X = 1CDA5B1212040420B77E8A268E0D3670371BBE7B44A7D8E5
Y = 30A74DDA9F952EFBAB4C3F72A3AA6BA98BE22636DDACC861

Changes

v1.9.1 beta

Suppressed the class number stuff.
The maximal value of the cofactor S can be specified (instead of its maximal binary size).
Over GF(P), the coefficients of a curve are in the range -P/2..P/2 whereas the coefficients of a point are in the range 0..P-1.
Help file entirely rewritten.

v1.0 beta 6

The lower bound 64 was decreased to 33 for both P binary size over GF(P) and field polynomial degree over GF(2ⁿ).

v1.0 beta 5

Help file updated.

v1.0 beta 4

Due to a very stupid bug (do sensible bugs exist?), the order of a curve over GF(2ⁿ) was sometimes set to 0 after having been computed.

v1.0 beta 3

Cosmetic bugs fixed.
The coefficient A of a curve over GF(P) may be random or any small integer in -10¹⁰..10¹⁰.