Merge pull request #302

03d4611 Add sage verification script for the group laws (Pieter Wuille)

sage/group_prover.sage

@@ -0,0 +1,322 @@
+# This code supports verifying group implementations which have branches
+# or conditional statements (like cmovs), by allowing each execution path
+# to independently set assumptions on input or intermediary variables.
+#
+# The general approach is:
+# * A constraint is a tuple of two sets of of symbolic expressions:
+#   the first of which are required to evaluate to zero, the second of which
+#   are required to evaluate to nonzero.
+#   - A constraint is said to be conflicting if any of its nonzero expressions
+#     is in the ideal with basis the zero expressions (in other words: when the
+#     zero expressions imply that one of the nonzero expressions are zero).
+# * There is a list of laws that describe the intended behaviour, including
+#   laws for addition and doubling. Each law is called with the symbolic point
+#   coordinates as arguments, and returns:
+#   - A constraint describing the assumptions under which it is applicable,
+#     called "assumeLaw"
+#   - A constraint describing the requirements of the law, called "require"
+# * Implementations are transliterated into functions that operate as well on
+#   algebraic input points, and are called once per combination of branches
+#   exectured. Each execution returns:
+#   - A constraint describing the assumptions this implementation requires
+#     (such as Z1=1), called "assumeFormula"
+#   - A constraint describing the assumptions this specific branch requires,
+#     but which is by construction guaranteed to cover the entire space by
+#     merging the results from all branches, called "assumeBranch"
+#   - The result of the computation
+# * All combinations of laws with implementation branches are tried, and:
+#   - If the combination of assumeLaw, assumeFormula, and assumeBranch results
+#     in a conflict, it means this law does not apply to this branch, and it is
+#     skipped.
+#   - For others, we try to prove the require constraints hold, assuming the
+#     information in assumeLaw + assumeFormula + assumeBranch, and if this does
+#     not succeed, we fail.
+#     + To prove an expression is zero, we check whether it belongs to the
+#       ideal with the assumed zero expressions as basis. This test is exact.
+#     + To prove an expression is nonzero, we check whether each of its
+#       factors is contained in the set of nonzero assumptions' factors.
+#       This test is not exact, so various combinations of original and
+#       reduced expressions' factors are tried.
+#   - If we succeed, we print out the assumptions from assumeFormula that
+#     weren't implied by assumeLaw already. Those from assumeBranch are skipped,
+#     as we assume that all constraints in it are complementary with each other.
+#
+# Based on the sage verification scripts used in the Explicit-Formulas Database
+# by Tanja Lange and others, see http://hyperelliptic.org/EFD
+
+class fastfrac:
+  """Fractions over rings."""
+
+  def __init__(self,R,top,bot=1):
+    """Construct a fractional, given a ring, a numerator, and denominator."""
+    self.R = R
+    if parent(top) == ZZ or parent(top) == R:
+      self.top = R(top)
+      self.bot = R(bot)
+    elif top.__class__ == fastfrac:
+      self.top = top.top
+      self.bot = top.bot * bot
+    else:
+      self.top = R(numerator(top))
+      self.bot = R(denominator(top)) * bot
+
+  def iszero(self,I):
+    """Return whether this fraction is zero given an ideal."""
+    return self.top in I and self.bot not in I
+
+  def reduce(self,assumeZero):
+    zero = self.R.ideal(map(numerator, assumeZero))
+    return fastfrac(self.R, zero.reduce(self.top)) / fastfrac(self.R, zero.reduce(self.bot))
+
+  def __add__(self,other):
+    """Add two fractions."""
+    if parent(other) == ZZ:
+      return fastfrac(self.R,self.top + self.bot * other,self.bot)
+    if other.__class__ == fastfrac:
+      return fastfrac(self.R,self.top * other.bot + self.bot * other.top,self.bot * other.bot)
+    return NotImplemented
+
+  def __sub__(self,other):
+    """Subtract two fractions."""
+    if parent(other) == ZZ:
+      return fastfrac(self.R,self.top - self.bot * other,self.bot)
+    if other.__class__ == fastfrac:
+      return fastfrac(self.R,self.top * other.bot - self.bot * other.top,self.bot * other.bot)
+    return NotImplemented
+
+  def __neg__(self):
+    """Return the negation of a fraction."""
+    return fastfrac(self.R,-self.top,self.bot)
+
+  def __mul__(self,other):
+    """Multiply two fractions."""
+    if parent(other) == ZZ:
+      return fastfrac(self.R,self.top * other,self.bot)
+    if other.__class__ == fastfrac:
+      return fastfrac(self.R,self.top * other.top,self.bot * other.bot)
+    return NotImplemented
+
+  def __rmul__(self,other):
+    """Multiply something else with a fraction."""
+    return self.__mul__(other)
+
+  def __div__(self,other):
+    """Divide two fractions."""
+    if parent(other) == ZZ:
+      return fastfrac(self.R,self.top,self.bot * other)
+    if other.__class__ == fastfrac:
+      return fastfrac(self.R,self.top * other.bot,self.bot * other.top)
+    return NotImplemented
+
+  def __pow__(self,other):
+    """Compute a power of a fraction."""
+    if parent(other) == ZZ:
+      if other < 0:
+        # Negative powers require flipping top and bottom
+        return fastfrac(self.R,self.bot ^ (-other),self.top ^ (-other))
+      else:
+        return fastfrac(self.R,self.top ^ other,self.bot ^ other)
+    return NotImplemented
+
+  def __str__(self):
+    return "fastfrac((" + str(self.top) + ") / (" + str(self.bot) + "))"
+  def __repr__(self):
+    return "%s" % self
+
+  def numerator(self):
+    return self.top
+
+class constraints:
+  """A set of constraints, consisting of zero and nonzero expressions.
+
+  Constraints can either be used to express knowledge or a requirement.
+
+  Both the fields zero and nonzero are maps from expressions to description
+  strings. The expressions that are the keys in zero are required to be zero,
+  and the expressions that are the keys in nonzero are required to be nonzero.
+
+  Note that (a != 0) and (b != 0) is the same as (a*b != 0), so all keys in
+  nonzero could be multiplied into a single key. This is often much less
+  efficient to work with though, so we keep them separate inside the
+  constraints. This allows higher-level code to do fast checks on the individual
+  nonzero elements, or combine them if needed for stronger checks.
+
+  We can't multiply the different zero elements, as it would suffice for one of
+  the factors to be zero, instead of all of them. Instead, the zero elements are
+  typically combined into an ideal first.
+  """
+
+  def __init__(self, **kwargs):
+    if 'zero' in kwargs:
+      self.zero = dict(kwargs['zero'])
+    else:
+      self.zero = dict()
+    if 'nonzero' in kwargs:
+      self.nonzero = dict(kwargs['nonzero'])
+    else:
+      self.nonzero = dict()
+
+  def negate(self):
+    return constraints(zero=self.nonzero, nonzero=self.zero)
+
+  def __add__(self, other):
+    zero = self.zero.copy()
+    zero.update(other.zero)
+    nonzero = self.nonzero.copy()
+    nonzero.update(other.nonzero)
+    return constraints(zero=zero, nonzero=nonzero)
+
+  def __str__(self):
+    return "constraints(zero=%s,nonzero=%s)" % (self.zero, self.nonzero)
+
+  def __repr__(self):
+    return "%s" % self
+
+
+def conflicts(R, con):
+  """Check whether any of the passed non-zero assumptions is implied by the zero assumptions"""
+  zero = R.ideal(map(numerator, con.zero))
+  if 1 in zero:
+    return True
+  # First a cheap check whether any of the individual nonzero terms conflict on
+  # their own.
+  for nonzero in con.nonzero:
+    if nonzero.iszero(zero):
+      return True
+  # It can be the case that entries in the nonzero set do not individually
+  # conflict with the zero set, but their combination does. For example, knowing
+  # that either x or y is zero is equivalent to having x*y in the zero set.
+  # Having x or y individually in the nonzero set is not a conflict, but both
+  # simultaneously is, so that is the right thing to check for.
+  if reduce(lambda a,b: a * b, con.nonzero, fastfrac(R, 1)).iszero(zero):
+    return True
+  return False
+
+
+def get_nonzero_set(R, assume):
+  """Calculate a simple set of nonzero expressions"""
+  zero = R.ideal(map(numerator, assume.zero))
+  nonzero = set()
+  for nz in map(numerator, assume.nonzero):
+    for (f,n) in nz.factor():
+      nonzero.add(f)
+    rnz = zero.reduce(nz)
+    for (f,n) in rnz.factor():
+      nonzero.add(f)
+  return nonzero
+
+
+def prove_nonzero(R, exprs, assume):
+  """Check whether an expression is provably nonzero, given assumptions"""
+  zero = R.ideal(map(numerator, assume.zero))
+  nonzero = get_nonzero_set(R, assume)
+  expl = set()
+  ok = True
+  for expr in exprs:
+    if numerator(expr) in zero:
+      return (False, [exprs[expr]])
+  allexprs = reduce(lambda a,b: numerator(a)*numerator(b), exprs, 1)
+  for (f, n) in allexprs.factor():
+    if f not in nonzero:
+      ok = False
+  if ok:
+    return (True, None)
+  ok = True
+  for (f, n) in zero.reduce(numerator(allexprs)).factor():
+    if f not in nonzero:
+      ok = False
+  if ok:
+    return (True, None)
+  ok = True
+  for expr in exprs:
+    for (f,n) in numerator(expr).factor():
+      if f not in nonzero:
+        ok = False
+  if ok:
+    return (True, None)
+  ok = True
+  for expr in exprs:
+    for (f,n) in zero.reduce(numerator(expr)).factor():
+      if f not in nonzero:
+        expl.add(exprs[expr])
+  if expl:
+    return (False, list(expl))
+  else:
+    return (True, None)
+
+
+def prove_zero(R, exprs, assume):
+  """Check whether all of the passed expressions are provably zero, given assumptions"""
+  r, e = prove_nonzero(R, dict(map(lambda x: (fastfrac(R, x.bot, 1), exprs[x]), exprs)), assume)
+  if not r:
+    return (False, map(lambda x: "Possibly zero denominator: %s" % x, e))
+  zero = R.ideal(map(numerator, assume.zero))
+  nonzero = prod(x for x in assume.nonzero)
+  expl = []
+  for expr in exprs:
+    if not expr.iszero(zero):
+      expl.append(exprs[expr])
+  if not expl:
+    return (True, None)
+  return (False, expl)
+
+
+def describe_extra(R, assume, assumeExtra):
+  """Describe what assumptions are added, given existing assumptions"""
+  zerox = assume.zero.copy()
+  zerox.update(assumeExtra.zero)
+  zero = R.ideal(map(numerator, assume.zero))
+  zeroextra = R.ideal(map(numerator, zerox))
+  nonzero = get_nonzero_set(R, assume)
+  ret = set()
+  # Iterate over the extra zero expressions
+  for base in assumeExtra.zero:
+    if base not in zero:
+      add = []
+      for (f, n) in numerator(base).factor():
+        if f not in nonzero:
+          add += ["%s" % f]
+      if add:
+        ret.add((" * ".join(add)) + " = 0 [%s]" % assumeExtra.zero[base])
+  # Iterate over the extra nonzero expressions
+  for nz in assumeExtra.nonzero:
+    nzr = zeroextra.reduce(numerator(nz))
+    if nzr not in zeroextra:
+      for (f,n) in nzr.factor():
+        if zeroextra.reduce(f) not in nonzero:
+          ret.add("%s != 0" % zeroextra.reduce(f))
+  return ", ".join(x for x in ret)
+
+
+def check_symbolic(R, assumeLaw, assumeAssert, assumeBranch, require):
+  """Check a set of zero and nonzero requirements, given a set of zero and nonzero assumptions"""
+  assume = assumeLaw + assumeAssert + assumeBranch
+
+  if conflicts(R, assume):
+    # This formula does not apply
+    return None
+
+  describe = describe_extra(R, assumeLaw + assumeBranch, assumeAssert)
+
+  ok, msg = prove_zero(R, require.zero, assume)
+  if not ok:
+    return "FAIL, %s fails (assuming %s)" % (str(msg), describe)
+
+  res, expl = prove_nonzero(R, require.nonzero, assume)
+  if not res:
+    return "FAIL, %s fails (assuming %s)" % (str(expl), describe)
+
+  if describe != "":
+    return "OK (assuming %s)" % describe
+  else:
+    return "OK"
+
+
+def concrete_verify(c):
+  for k in c.zero:
+    if k != 0:
+      return (False, c.zero[k])
+  for k in c.nonzero:
+    if k == 0:
+      return (False, c.nonzero[k])
+  return (True, None)

sage/secp256k1.sage

@@ -0,0 +1,306 @@
+# 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)

sage/weierstrass_prover.sage

@@ -0,0 +1,264 @@
+# 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