ark-ec 0.5.0

A library for elliptic curves and pairings
Documentation 
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# Arguments:
# - F, a field object, e.g., F = GF(2^521 - 1)
# - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
def find_z_sswu(F, A, B):
    R.<xx> = F[]
    # Polynomial ring over F
    g = xx^3 + F(A) * xx + F(B)
    # y^2 = g(x) = x^3 + A * x + B
    ctr = F.gen()
    while True:
        for Z_cand in (F(ctr), F(-ctr)):
            # Criterion 1: Z is non-square in F.
            if is_square(Z_cand):
                continue
            # Criterion 2: Z != -1 in F.
            if Z_cand == F(-1):
                continue
            # Criterion 3: g(x) - Z is irreducible over F.
            if not (g - Z_cand).is_irreducible():
                continue
            # Criterion 4: g(B / (Z * A)) is square in F.
            if is_square(g(B / (Z_cand * A))):
                return Z_cand
        ctr += 1

# Finds the smallest z in term of non-zero bit
# in sage representation for constructing
# elligator2 map for a curve defined over field F.
# Argument:
# - F, a field object, e.g., F = GF(2^255 - 19)
def find_z_ell2(F):
    ctr = F.gen()
    while True:
        for Z_cand in (F(ctr), F(-ctr)):
            # Z must be a non-square in F.
            if is_square(Z_cand):
                continue
            return Z_cand
        ctr += 1