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Diffstat (limited to 'crypto/secp256k1/libsecp256k1/sage/group_prover.sage')
| -rw-r--r-- | crypto/secp256k1/libsecp256k1/sage/group_prover.sage | 322 |
1 files changed, 0 insertions, 322 deletions
diff --git a/crypto/secp256k1/libsecp256k1/sage/group_prover.sage b/crypto/secp256k1/libsecp256k1/sage/group_prover.sage deleted file mode 100644 index ab580c5b2..000000000 --- a/crypto/secp256k1/libsecp256k1/sage/group_prover.sage +++ /dev/null @@ -1,322 +0,0 @@ -# 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) |