307 lines
8.6 KiB
Python
307 lines
8.6 KiB
Python
|
# Test libsecp256k1' group operation implementations using prover.sage
|
||
|
|
|
||
|
|
import sys
|
||
|
|
|
||
|
|
load("group_prover.sage")
|
||
|
|
load("weierstrass_prover.sage")
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_double_var(a):
|
||
|
|
"""libsecp256k1's secp256k1_gej_double_var, used by various addition functions"""
|
||
|
|
rz = a.Z * a.Y
|
||
|
|
rz = rz * 2
|
||
|
|
t1 = a.X^2
|
||
|
|
t1 = t1 * 3
|
||
|
|
t2 = t1^2
|
||
|
|
t3 = a.Y^2
|
||
|
|
t3 = t3 * 2
|
||
|
|
t4 = t3^2
|
||
|
|
t4 = t4 * 2
|
||
|
|
t3 = t3 * a.X
|
||
|
|
rx = t3
|
||
|
|
rx = rx * 4
|
||
|
|
rx = -rx
|
||
|
|
rx = rx + t2
|
||
|
|
t2 = -t2
|
||
|
|
t3 = t3 * 6
|
||
|
|
t3 = t3 + t2
|
||
|
|
ry = t1 * t3
|
||
|
|
t2 = -t4
|
||
|
|
ry = ry + t2
|
||
|
|
return jacobianpoint(rx, ry, rz)
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_add_var(branch, a, b):
|
||
|
|
"""libsecp256k1's secp256k1_gej_add_var"""
|
||
|
|
if branch == 0:
|
||
|
|
return (constraints(), constraints(nonzero={a.Infinity : 'a_infinite'}), b)
|
||
|
|
if branch == 1:
|
||
|
|
return (constraints(), constraints(zero={a.Infinity : 'a_finite'}, nonzero={b.Infinity : 'b_infinite'}), a)
|
||
|
|
z22 = b.Z^2
|
||
|
|
z12 = a.Z^2
|
||
|
|
u1 = a.X * z22
|
||
|
|
u2 = b.X * z12
|
||
|
|
s1 = a.Y * z22
|
||
|
|
s1 = s1 * b.Z
|
||
|
|
s2 = b.Y * z12
|
||
|
|
s2 = s2 * a.Z
|
||
|
|
h = -u1
|
||
|
|
h = h + u2
|
||
|
|
i = -s1
|
||
|
|
i = i + s2
|
||
|
|
if branch == 2:
|
||
|
|
r = formula_secp256k1_gej_double_var(a)
|
||
|
|
return (constraints(), constraints(zero={h : 'h=0', i : 'i=0', a.Infinity : 'a_finite', b.Infinity : 'b_finite'}), r)
|
||
|
|
if branch == 3:
|
||
|
|
return (constraints(), constraints(zero={h : 'h=0', a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={i : 'i!=0'}), point_at_infinity())
|
||
|
|
i2 = i^2
|
||
|
|
h2 = h^2
|
||
|
|
h3 = h2 * h
|
||
|
|
h = h * b.Z
|
||
|
|
rz = a.Z * h
|
||
|
|
t = u1 * h2
|
||
|
|
rx = t
|
||
|
|
rx = rx * 2
|
||
|
|
rx = rx + h3
|
||
|
|
rx = -rx
|
||
|
|
rx = rx + i2
|
||
|
|
ry = -rx
|
||
|
|
ry = ry + t
|
||
|
|
ry = ry * i
|
||
|
|
h3 = h3 * s1
|
||
|
|
h3 = -h3
|
||
|
|
ry = ry + h3
|
||
|
|
return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_add_ge_var(branch, a, b):
|
||
|
|
"""libsecp256k1's secp256k1_gej_add_ge_var, which assume bz==1"""
|
||
|
|
if branch == 0:
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(nonzero={a.Infinity : 'a_infinite'}), b)
|
||
|
|
if branch == 1:
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite'}, nonzero={b.Infinity : 'b_infinite'}), a)
|
||
|
|
z12 = a.Z^2
|
||
|
|
u1 = a.X
|
||
|
|
u2 = b.X * z12
|
||
|
|
s1 = a.Y
|
||
|
|
s2 = b.Y * z12
|
||
|
|
s2 = s2 * a.Z
|
||
|
|
h = -u1
|
||
|
|
h = h + u2
|
||
|
|
i = -s1
|
||
|
|
i = i + s2
|
||
|
|
if (branch == 2):
|
||
|
|
r = formula_secp256k1_gej_double_var(a)
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0', i : 'i=0'}), r)
|
||
|
|
if (branch == 3):
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0'}, nonzero={i : 'i!=0'}), point_at_infinity())
|
||
|
|
i2 = i^2
|
||
|
|
h2 = h^2
|
||
|
|
h3 = h * h2
|
||
|
|
rz = a.Z * h
|
||
|
|
t = u1 * h2
|
||
|
|
rx = t
|
||
|
|
rx = rx * 2
|
||
|
|
rx = rx + h3
|
||
|
|
rx = -rx
|
||
|
|
rx = rx + i2
|
||
|
|
ry = -rx
|
||
|
|
ry = ry + t
|
||
|
|
ry = ry * i
|
||
|
|
h3 = h3 * s1
|
||
|
|
h3 = -h3
|
||
|
|
ry = ry + h3
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_add_zinv_var(branch, a, b):
|
||
|
|
"""libsecp256k1's secp256k1_gej_add_zinv_var"""
|
||
|
|
bzinv = b.Z^(-1)
|
||
|
|
if branch == 0:
|
||
|
|
return (constraints(), constraints(nonzero={b.Infinity : 'b_infinite'}), a)
|
||
|
|
if branch == 1:
|
||
|
|
bzinv2 = bzinv^2
|
||
|
|
bzinv3 = bzinv2 * bzinv
|
||
|
|
rx = b.X * bzinv2
|
||
|
|
ry = b.Y * bzinv3
|
||
|
|
rz = 1
|
||
|
|
return (constraints(), constraints(zero={b.Infinity : 'b_finite'}, nonzero={a.Infinity : 'a_infinite'}), jacobianpoint(rx, ry, rz))
|
||
|
|
azz = a.Z * bzinv
|
||
|
|
z12 = azz^2
|
||
|
|
u1 = a.X
|
||
|
|
u2 = b.X * z12
|
||
|
|
s1 = a.Y
|
||
|
|
s2 = b.Y * z12
|
||
|
|
s2 = s2 * azz
|
||
|
|
h = -u1
|
||
|
|
h = h + u2
|
||
|
|
i = -s1
|
||
|
|
i = i + s2
|
||
|
|
if branch == 2:
|
||
|
|
r = formula_secp256k1_gej_double_var(a)
|
||
|
|
return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0', i : 'i=0'}), r)
|
||
|
|
if branch == 3:
|
||
|
|
return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0'}, nonzero={i : 'i!=0'}), point_at_infinity())
|
||
|
|
i2 = i^2
|
||
|
|
h2 = h^2
|
||
|
|
h3 = h * h2
|
||
|
|
rz = a.Z
|
||
|
|
rz = rz * h
|
||
|
|
t = u1 * h2
|
||
|
|
rx = t
|
||
|
|
rx = rx * 2
|
||
|
|
rx = rx + h3
|
||
|
|
rx = -rx
|
||
|
|
rx = rx + i2
|
||
|
|
ry = -rx
|
||
|
|
ry = ry + t
|
||
|
|
ry = ry * i
|
||
|
|
h3 = h3 * s1
|
||
|
|
h3 = -h3
|
||
|
|
ry = ry + h3
|
||
|
|
return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_add_ge(branch, a, b):
|
||
|
|
"""libsecp256k1's secp256k1_gej_add_ge"""
|
||
|
|
zeroes = {}
|
||
|
|
nonzeroes = {}
|
||
|
|
a_infinity = False
|
||
|
|
if (branch & 4) != 0:
|
||
|
|
nonzeroes.update({a.Infinity : 'a_infinite'})
|
||
|
|
a_infinity = True
|
||
|
|
else:
|
||
|
|
zeroes.update({a.Infinity : 'a_finite'})
|
||
|
|
zz = a.Z^2
|
||
|
|
u1 = a.X
|
||
|
|
u2 = b.X * zz
|
||
|
|
s1 = a.Y
|
||
|
|
s2 = b.Y * zz
|
||
|
|
s2 = s2 * a.Z
|
||
|
|
t = u1
|
||
|
|
t = t + u2
|
||
|
|
m = s1
|
||
|
|
m = m + s2
|
||
|
|
rr = t^2
|
||
|
|
m_alt = -u2
|
||
|
|
tt = u1 * m_alt
|
||
|
|
rr = rr + tt
|
||
|
|
degenerate = (branch & 3) == 3
|
||
|
|
if (branch & 1) != 0:
|
||
|
|
zeroes.update({m : 'm_zero'})
|
||
|
|
else:
|
||
|
|
nonzeroes.update({m : 'm_nonzero'})
|
||
|
|
if (branch & 2) != 0:
|
||
|
|
zeroes.update({rr : 'rr_zero'})
|
||
|
|
else:
|
||
|
|
nonzeroes.update({rr : 'rr_nonzero'})
|
||
|
|
rr_alt = s1
|
||
|
|
rr_alt = rr_alt * 2
|
||
|
|
m_alt = m_alt + u1
|
||
|
|
if not degenerate:
|
||
|
|
rr_alt = rr
|
||
|
|
m_alt = m
|
||
|
|
n = m_alt^2
|
||
|
|
q = n * t
|
||
|
|
n = n^2
|
||
|
|
if degenerate:
|
||
|
|
n = m
|
||
|
|
t = rr_alt^2
|
||
|
|
rz = a.Z * m_alt
|
||
|
|
infinity = False
|
||
|
|
if (branch & 8) != 0:
|
||
|
|
if not a_infinity:
|
||
|
|
infinity = True
|
||
|
|
zeroes.update({rz : 'r.z=0'})
|
||
|
|
else:
|
||
|
|
nonzeroes.update({rz : 'r.z!=0'})
|
||
|
|
rz = rz * 2
|
||
|
|
q = -q
|
||
|
|
t = t + q
|
||
|
|
rx = t
|
||
|
|
t = t * 2
|
||
|
|
t = t + q
|
||
|
|
t = t * rr_alt
|
||
|
|
t = t + n
|
||
|
|
ry = -t
|
||
|
|
rx = rx * 4
|
||
|
|
ry = ry * 4
|
||
|
|
if a_infinity:
|
||
|
|
rx = b.X
|
||
|
|
ry = b.Y
|
||
|
|
rz = 1
|
||
|
|
if infinity:
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zeroes, nonzero=nonzeroes), point_at_infinity())
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zeroes, nonzero=nonzeroes), jacobianpoint(rx, ry, rz))
|
||
|
|
|
||
|
|
def formula_secp256k1_gej_add_ge_old(branch, a, b):
|
||
|
|
"""libsecp256k1's old secp256k1_gej_add_ge, which fails when ay+by=0 but ax!=bx"""
|
||
|
|
a_infinity = (branch & 1) != 0
|
||
|
|
zero = {}
|
||
|
|
nonzero = {}
|
||
|
|
if a_infinity:
|
||
|
|
nonzero.update({a.Infinity : 'a_infinite'})
|
||
|
|
else:
|
||
|
|
zero.update({a.Infinity : 'a_finite'})
|
||
|
|
zz = a.Z^2
|
||
|
|
u1 = a.X
|
||
|
|
u2 = b.X * zz
|
||
|
|
s1 = a.Y
|
||
|
|
s2 = b.Y * zz
|
||
|
|
s2 = s2 * a.Z
|
||
|
|
z = a.Z
|
||
|
|
t = u1
|
||
|
|
t = t + u2
|
||
|
|
m = s1
|
||
|
|
m = m + s2
|
||
|
|
n = m^2
|
||
|
|
q = n * t
|
||
|
|
n = n^2
|
||
|
|
rr = t^2
|
||
|
|
t = u1 * u2
|
||
|
|
t = -t
|
||
|
|
rr = rr + t
|
||
|
|
t = rr^2
|
||
|
|
rz = m * z
|
||
|
|
infinity = False
|
||
|
|
if (branch & 2) != 0:
|
||
|
|
if not a_infinity:
|
||
|
|
infinity = True
|
||
|
|
else:
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(nonzero={z : 'conflict_a'}, zero={z : 'conflict_b'}), point_at_infinity())
|
||
|
|
zero.update({rz : 'r.z=0'})
|
||
|
|
else:
|
||
|
|
nonzero.update({rz : 'r.z!=0'})
|
||
|
|
rz = rz * (0 if a_infinity else 2)
|
||
|
|
rx = t
|
||
|
|
q = -q
|
||
|
|
rx = rx + q
|
||
|
|
q = q * 3
|
||
|
|
t = t * 2
|
||
|
|
t = t + q
|
||
|
|
t = t * rr
|
||
|
|
t = t + n
|
||
|
|
ry = -t
|
||
|
|
rx = rx * (0 if a_infinity else 4)
|
||
|
|
ry = ry * (0 if a_infinity else 4)
|
||
|
|
t = b.X
|
||
|
|
t = t * (1 if a_infinity else 0)
|
||
|
|
rx = rx + t
|
||
|
|
t = b.Y
|
||
|
|
t = t * (1 if a_infinity else 0)
|
||
|
|
ry = ry + t
|
||
|
|
t = (1 if a_infinity else 0)
|
||
|
|
rz = rz + t
|
||
|
|
if infinity:
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zero, nonzero=nonzero), point_at_infinity())
|
||
|
|
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zero, nonzero=nonzero), jacobianpoint(rx, ry, rz))
|
||
|
|
|
||
|
|
if __name__ == "__main__":
|
||
|
|
check_symbolic_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var)
|
||
|
|
check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var)
|
||
|
|
check_symbolic_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var)
|
||
|
|
check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge)
|
||
|
|
check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old)
|
||
|
|
|
||
|
|
if len(sys.argv) >= 2 and sys.argv[1] == "--exhaustive":
|
||
|
|
check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var, 43)
|
||
|
|
check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var, 43)
|
||
|
|
check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var, 43)
|
||
|
|
check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge, 43)
|
||
|
|
check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old, 43)
|