so any point should work as a G.
Ah ha! Thanks. I was wondering why the group order needed to be prime.
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sage: F = FiniteField(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
sage: F
Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: C = EllipticCurve(F, [ 0, 7 ])
sage: C
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
sage: print(C.cardinality())
115792089237316195423570985008687907852837564279074904382605163141518161494337
sage: hex(C.cardinality())
'fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141'
sage: G = C.point((0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798, 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8))
sage: G
(55066263022277343669578718895168534326250603453777594175500187360389116729240 : 32670510020758816978083085130507043184471273380659243275938904335757337482424 : 1)
sage: G.order()
115792089237316195423570985008687907852837564279074904382605163141518161494337
sage: hex(G.order())
'fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141'
from sage.all import *
def find_group (prime):
K = FiniteField (prime)
for b in xrange (1, 1000):
E = EllipticCurve (K, [0, b])
if E.order () in Primes ():
return b, E.order ()
return None
p = 2**256 - 2**32 + 1
while True:
p -= 1
if not p in Primes ():
continue
result = find_group (p)
if result is not None:
break
print "Found curve: "
print "p = %X" % p
print "a = 0"
print "b = %d" % result[0]
print "N = %X" % result[1]