323 lines
12 KiB
Python
323 lines
12 KiB
Python
# 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 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
|
|
# executed. 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)
|