siec

Super-Isolated Elliptic Curve Implementation in Go

This package exports a super-isolated elliptic curve. Over the base field 𝔽ₚ, the curve E does not admit any isogenies to other curves.

We can verify the curve properties in Sage.

K.<isqrt3> = QuadraticField(-3)
pi = $2^{127}$ + $2^{25}$ + $2^{12}$ + $2^{6}$ + (1 - isqrt3)/2
p = ZZ(pi.norm())
N = ZZ((pi-1).norm())
E = EllipticCurve(GF(p),[0,19]) # E: y^2 = x^3 + 19
G = E([5,12])
# p is a 255 bit prime with hamming weight 14
assert p.is_prime()
assert len(p.bits()) == 255
assert sum(p.bits()) == 14
# N is a 255 bit prime
assert N.is_prime()
assert len(N.bits()) == 255
# E has N points
assert E.count_points() == N
# The Frobenius endomorphism on E satisfies the same minimal polynomial as pi
assert E.frobenius_polynomial() == pi.minpoly()
# pi generates a maximal order
assert K.order([pi]).is_maximal()
# K has class number 1
assert K.class_number() == 1
# Examples of order 6 endomorphisms
assert E([28948022309329048855892746252183396360433790236562615305258360005404201062400*G[0],-G[1]]) == 170141183460469231731687303715917664320*G
assert $170141183460469231731687303715917664320^{3}$ % N == N-1