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prove_group_implementations.sage
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prove_group_implementations.sage
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# 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
s = a.Y^2
l = a.X^2
l = l * 3
l = l / 2
t = -s
t = t * a.X
rx = l^2
rx = rx + t
rx = rx + t
s = s^2
t = t + rx
ry = t * l
ry = ry + s
ry = -ry
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 & 2) != 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 & 1) != 0
if degenerate:
zeroes.update({m : 'm_zero'})
else:
nonzeroes.update({m : 'm_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 = -t
q = q * n
n = n^2
if degenerate:
n = m
t = rr_alt^2
rz = a.Z * m_alt
t = t + q
rx = t
t = t * 2
t = t + q
t = t * rr_alt
t = t + n
ry = -t
ry = ry / 2
if a_infinity:
rx = b.X
ry = b.Y
rz = 1
if (branch & 4) != 0:
zeroes.update({rz : 'r.z = 0'})
return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zeroes, nonzero=nonzeroes), point_at_infinity())
else:
nonzeroes.update({rz : 'r.z != 0'})
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__":
success = True
success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var)
success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var)
success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var)
success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 8, formula_secp256k1_gej_add_ge)
success = success & (not 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":
success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var, 43)
success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var, 43)
success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var, 43)
success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 8, formula_secp256k1_gej_add_ge, 43)
success = success & (not check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old, 43))
sys.exit(int(not success))