git / moonbite-core / src/secp256k1-zkp/sage/weierstrass_prover.sage · raw
# Prover implementation for Weierstrass curves of the form# y^2 = x^3 + A * x + B, specifically with a = 0 and b = 7, with group laws# operating on affine and Jacobian coordinates, including the point at infinity# represented by a 4th variable in coordinates.load("group_prover.sage")class affinepoint: def __init__(self, x, y, infinity=0): self.x = x self.y = y self.infinity = infinity def __str__(self): return "affinepoint(x=%s,y=%s,inf=%s)" % (self.x, self.y, self.infinity)class jacobianpoint: def __init__(self, x, y, z, infinity=0): self.X = x self.Y = y self.Z = z self.Infinity = infinity def __str__(self): return "jacobianpoint(X=%s,Y=%s,Z=%s,inf=%s)" % (self.X, self.Y, self.Z, self.Infinity)def point_at_infinity(): return jacobianpoint(1, 1, 1, 1)def negate(p): if p.__class__ == affinepoint: return affinepoint(p.x, -p.y) if p.__class__ == jacobianpoint: return jacobianpoint(p.X, -p.Y, p.Z) assert(False)def on_weierstrass_curve(A, B, p): """Return a set of zero-expressions for an affine point to be on the curve""" return constraints(zero={p.x^3 + A*p.x + B - p.y^2: 'on_curve'})def tangential_to_weierstrass_curve(A, B, p12, p3): """Return a set of zero-expressions for ((x12,y12),(x3,y3)) to be a line that is tangential to the curve at (x12,y12)""" return constraints(zero={ (p12.y - p3.y) * (p12.y * 2) - (p12.x^2 * 3 + A) * (p12.x - p3.x): 'tangential_to_curve' })def colinear(p1, p2, p3): """Return a set of zero-expressions for ((x1,y1),(x2,y2),(x3,y3)) to be collinear""" return constraints(zero={ (p1.y - p2.y) * (p1.x - p3.x) - (p1.y - p3.y) * (p1.x - p2.x): 'colinear_1', (p2.y - p3.y) * (p2.x - p1.x) - (p2.y - p1.y) * (p2.x - p3.x): 'colinear_2', (p3.y - p1.y) * (p3.x - p2.x) - (p3.y - p2.y) * (p3.x - p1.x): 'colinear_3' })def good_affine_point(p): return constraints(nonzero={p.x : 'nonzero_x', p.y : 'nonzero_y'})def good_jacobian_point(p): return constraints(nonzero={p.X : 'nonzero_X', p.Y : 'nonzero_Y', p.Z^6 : 'nonzero_Z'})def good_point(p): return constraints(nonzero={p.Z^6 : 'nonzero_X'})def finite(p, *affine_fns): con = good_point(p) + constraints(zero={p.Infinity : 'finite_point'}) if p.Z != 0: return con + reduce(lambda a, b: a + b, (f(affinepoint(p.X / p.Z^2, p.Y / p.Z^3)) for f in affine_fns), con) else: return condef infinite(p): return constraints(nonzero={p.Infinity : 'infinite_point'})def law_jacobian_weierstrass_add(A, B, pa, pb, pA, pB, pC): """Check whether the passed set of coordinates is a valid Jacobian add, given assumptions""" assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + on_weierstrass_curve(A, B, pa) + on_weierstrass_curve(A, B, pb) + finite(pA) + finite(pB) + constraints(nonzero={pa.x - pb.x : 'different_x'})) require = (finite(pC, lambda pc: on_weierstrass_curve(A, B, pc) + colinear(pa, pb, negate(pc)))) return (assumeLaw, require)def law_jacobian_weierstrass_double(A, B, pa, pb, pA, pB, pC): """Check whether the passed set of coordinates is a valid Jacobian doubling, given assumptions""" assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + on_weierstrass_curve(A, B, pa) + on_weierstrass_curve(A, B, pb) + finite(pA) + finite(pB) + constraints(zero={pa.x - pb.x : 'equal_x', pa.y - pb.y : 'equal_y'})) require = (finite(pC, lambda pc: on_weierstrass_curve(A, B, pc) + tangential_to_weierstrass_curve(A, B, pa, negate(pc)))) return (assumeLaw, require)def law_jacobian_weierstrass_add_opposites(A, B, pa, pb, pA, pB, pC): assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + on_weierstrass_curve(A, B, pa) + on_weierstrass_curve(A, B, pb) + finite(pA) + finite(pB) + constraints(zero={pa.x - pb.x : 'equal_x', pa.y + pb.y : 'opposite_y'})) require = infinite(pC) return (assumeLaw, require)def law_jacobian_weierstrass_add_infinite_a(A, B, pa, pb, pA, pB, pC): assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + on_weierstrass_curve(A, B, pb) + infinite(pA) + finite(pB)) require = finite(pC, lambda pc: constraints(zero={pc.x - pb.x : 'c.x=b.x', pc.y - pb.y : 'c.y=b.y'})) return (assumeLaw, require)def law_jacobian_weierstrass_add_infinite_b(A, B, pa, pb, pA, pB, pC): assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + on_weierstrass_curve(A, B, pa) + infinite(pB) + finite(pA)) require = finite(pC, lambda pc: constraints(zero={pc.x - pa.x : 'c.x=a.x', pc.y - pa.y : 'c.y=a.y'})) return (assumeLaw, require)def law_jacobian_weierstrass_add_infinite_ab(A, B, pa, pb, pA, pB, pC): assumeLaw = (good_affine_point(pa) + good_affine_point(pb) + good_jacobian_point(pA) + good_jacobian_point(pB) + infinite(pA) + infinite(pB)) require = infinite(pC) return (assumeLaw, require)laws_jacobian_weierstrass = { 'add': law_jacobian_weierstrass_add, 'double': law_jacobian_weierstrass_double, 'add_opposite': law_jacobian_weierstrass_add_opposites, 'add_infinite_a': law_jacobian_weierstrass_add_infinite_a, 'add_infinite_b': law_jacobian_weierstrass_add_infinite_b, 'add_infinite_ab': law_jacobian_weierstrass_add_infinite_ab}def check_exhaustive_jacobian_weierstrass(name, A, B, branches, formula, p): """Verify an implementation of addition of Jacobian points on a Weierstrass curve, by executing and validating the result for every possible addition in a prime field""" F = Integers(p) print "Formula %s on Z%i:" % (name, p) points = [] for x in xrange(0, p): for y in xrange(0, p): point = affinepoint(F(x), F(y)) r, e = concrete_verify(on_weierstrass_curve(A, B, point)) if r: points.append(point) for za in xrange(1, p): for zb in xrange(1, p): for pa in points: for pb in points: for ia in xrange(2): for ib in xrange(2): pA = jacobianpoint(pa.x * F(za)^2, pa.y * F(za)^3, F(za), ia) pB = jacobianpoint(pb.x * F(zb)^2, pb.y * F(zb)^3, F(zb), ib) for branch in xrange(0, branches): assumeAssert, assumeBranch, pC = formula(branch, pA, pB) pC.X = F(pC.X) pC.Y = F(pC.Y) pC.Z = F(pC.Z) pC.Infinity = F(pC.Infinity) r, e = concrete_verify(assumeAssert + assumeBranch) if r: match = False for key in laws_jacobian_weierstrass: assumeLaw, require = laws_jacobian_weierstrass[key](A, B, pa, pb, pA, pB, pC) r, e = concrete_verify(assumeLaw) if r: if match: print " multiple branches for (%s,%s,%s,%s) + (%s,%s,%s,%s)" % (pA.X, pA.Y, pA.Z, pA.Infinity, pB.X, pB.Y, pB.Z, pB.Infinity) else: match = True r, e = concrete_verify(require) if not r: print " failure in branch %i for (%s,%s,%s,%s) + (%s,%s,%s,%s) = (%s,%s,%s,%s): %s" % (branch, pA.X, pA.Y, pA.Z, pA.Infinity, pB.X, pB.Y, pB.Z, pB.Infinity, pC.X, pC.Y, pC.Z, pC.Infinity, e) printdef check_symbolic_function(R, assumeAssert, assumeBranch, f, A, B, pa, pb, pA, pB, pC): assumeLaw, require = f(A, B, pa, pb, pA, pB, pC) return check_symbolic(R, assumeLaw, assumeAssert, assumeBranch, require)def check_symbolic_jacobian_weierstrass(name, A, B, branches, formula): """Verify an implementation of addition of Jacobian points on a Weierstrass curve symbolically""" R.<ax,bx,ay,by,Az,Bz,Ai,Bi> = PolynomialRing(QQ,8,order='invlex') lift = lambda x: fastfrac(R,x) ax = lift(ax) ay = lift(ay) Az = lift(Az) bx = lift(bx) by = lift(by) Bz = lift(Bz) Ai = lift(Ai) Bi = lift(Bi) pa = affinepoint(ax, ay, Ai) pb = affinepoint(bx, by, Bi) pA = jacobianpoint(ax * Az^2, ay * Az^3, Az, Ai) pB = jacobianpoint(bx * Bz^2, by * Bz^3, Bz, Bi) res = {} for key in laws_jacobian_weierstrass: res[key] = [] print ("Formula " + name + ":") count = 0 for branch in xrange(branches): assumeFormula, assumeBranch, pC = formula(branch, pA, pB) pC.X = lift(pC.X) pC.Y = lift(pC.Y) pC.Z = lift(pC.Z) pC.Infinity = lift(pC.Infinity) for key in laws_jacobian_weierstrass: res[key].append((check_symbolic_function(R, assumeFormula, assumeBranch, laws_jacobian_weierstrass[key], A, B, pa, pb, pA, pB, pC), branch)) for key in res: print " %s:" % key val = res[key] for x in val: if x[0] is not None: print " branch %i: %s" % (x[1], x[0]) print