mirror of
https://github.com/hashcat/hashcat.git
synced 2024-12-30 18:31:00 +00:00
265 lines
9.3 KiB
Python
265 lines
9.3 KiB
Python
# 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 con
|
|
|
|
def 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)
|
|
print
|
|
|
|
|
|
def 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
|