SAT Functions for Boolean Polynomials#
These highlevel functions support solving and learning from Boolean polynomial systems. In this context, “learning” means the construction of new polynomials in the ideal spanned by the original polynomials.
AUTHOR:
Martin Albrecht (2012): initial version
Functions#
- sage.sat.boolean_polynomials.learn(F, converter=None, solver=None, max_learnt_length=3, interreduction=False, **kwds)#
Learn new polynomials by running SAT-solver
solveron SAT-instance produced byconverterfromF.INPUT:
F- a sequence of Boolean polynomialsconverter- an ANF to CNF converter class or object. IfconverterisNonethensage.sat.converters.polybori.CNFEncoderis used to construct a new converter. (default:None)solver- a SAT-solver class or object. IfsolverisNonethensage.sat.solvers.cryptominisat.CryptoMiniSatis used to construct a new converter. (default:None)max_learnt_length- only clauses of length <=max_length_learntare considered and converted to polynomials. (default:3)interreduction- inter-reduce the resulting polynomials (default:False)
Note
More parameters can be passed to the converter and the solver by prefixing them with
c_ands_respectively. For example, to increase CryptoMiniSat’s verbosity level, passs_verbosity=1.OUTPUT:
A sequence of Boolean polynomials.
EXAMPLES:
sage: from sage.sat.boolean_polynomials import learn as learn_sat # optional - pycryptosat
We construct a simple system and solve it:
sage: set_random_seed(2300) # optional - pycryptosat sage: sr = mq.SR(1,2,2,4,gf2=True,polybori=True) # optional - pycryptosat sage: F,s = sr.polynomial_system() # optional - pycryptosat sage: H = learn_sat(F) # optional - pycryptosat sage: H[-1] # optional - pycryptosat k033 + 1
- sage.sat.boolean_polynomials.solve(F, converter=None, solver=None, n=1, target_variables=None, **kwds)#
Solve system of Boolean polynomials
Fby solving the SAT-problem – produced byconverter– usingsolver.INPUT:
F- a sequence of Boolean polynomialsn- number of solutions to return. Ifnis +infinity then all solutions are returned. Ifn <infinitythennsolutions are returned ifFhas at leastnsolutions. Otherwise, all solutions ofFare returned. (default:1)converter- an ANF to CNF converter class or object. IfconverterisNonethensage.sat.converters.polybori.CNFEncoderis used to construct a new converter. (default:None)solver- a SAT-solver class or object. IfsolverisNonethensage.sat.solvers.cryptominisat.CryptoMiniSatis used to construct a new converter. (default:None)target_variables- a list of variables. The elements of the list are used to exclude a particular combination of variable assignments of a solution from any further solution. Furthermoretarget_variablesdenotes which variable-value pairs appear in the solutions. Iftarget_variablesisNoneall variables appearing in the polynomials ofFare used to construct exclusion clauses. (default:None)**kwds- parameters can be passed to the converter and thesolver by prefixing them with
c_ands_respectively. For example, to increase CryptoMiniSat’s verbosity level, passs_verbosity=1.
OUTPUT:
A list of dictionaries, each of which contains a variable assignment solving
F.EXAMPLES:
We construct a very small-scale AES system of equations:
sage: sr = mq.SR(1,1,1,4,gf2=True,polybori=True) sage: while True: # workaround (see :trac:`31891`) ....: try: ....: F, s = sr.polynomial_system() ....: break ....: except ZeroDivisionError: ....: pass
and pass it to a SAT solver:
sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - pycryptosat sage: s = solve_sat(F) # optional - pycryptosat sage: F.subs(s[0]) # optional - pycryptosat Polynomial Sequence with 36 Polynomials in 0 Variables
This time we pass a few options through to the converter and the solver:
sage: s = solve_sat(F, c_max_vars_sparse=4, c_cutting_number=8) # optional - pycryptosat sage: F.subs(s[0]) # optional - pycryptosat Polynomial Sequence with 36 Polynomials in 0 Variables
We construct a very simple system with three solutions and ask for a specific number of solutions:
sage: B.<a,b> = BooleanPolynomialRing() # optional - pycryptosat sage: f = a*b # optional - pycryptosat sage: l = solve_sat([f],n=1) # optional - pycryptosat sage: len(l) == 1, f.subs(l[0]) # optional - pycryptosat (True, 0) sage: l = solve_sat([a*b],n=2) # optional - pycryptosat sage: len(l) == 2, f.subs(l[0]), f.subs(l[1]) # optional - pycryptosat (True, 0, 0) sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=3)) # optional - pycryptosat [(0, 0), (0, 1), (1, 0)] sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=4)) # optional - pycryptosat [(0, 0), (0, 1), (1, 0)] sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=infinity)) # optional - pycryptosat [(0, 0), (0, 1), (1, 0)]
In the next example we see how the
target_variablesparameter works:sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - pycryptosat sage: R.<a,b,c,d> = BooleanPolynomialRing() # optional - pycryptosat sage: F = [a+b,a+c+d] # optional - pycryptosat
First the normal use case:
sage: sorted((D[a], D[b], D[c], D[d]) for D in solve_sat(F,n=infinity)) # optional - pycryptosat [(0, 0, 0, 0), (0, 0, 1, 1), (1, 1, 0, 1), (1, 1, 1, 0)]
Now we are only interested in the solutions of the variables a and b:
sage: solve_sat(F,n=infinity,target_variables=[a,b]) # optional - pycryptosat [{b: 0, a: 0}, {b: 1, a: 1}]
Here, we generate and solve the cubic equations of the AES SBox (see trac ticket #26676):
sage: from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence # optional - pycryptosat, long time sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - pycryptosat, long time sage: sr = sage.crypto.mq.SR(1, 4, 4, 8, allow_zero_inversions = True) # optional - pycryptosat, long time sage: sb = sr.sbox() # optional - pycryptosat, long time sage: eqs = sb.polynomials(degree = 3) # optional - pycryptosat, long time sage: eqs = PolynomialSequence(eqs) # optional - pycryptosat, long time sage: variables = map(str, eqs.variables()) # optional - pycryptosat, long time sage: variables = ",".join(variables) # optional - pycryptosat, long time sage: R = BooleanPolynomialRing(16, variables) # optional - pycryptosat, long time sage: eqs = [R(eq) for eq in eqs] # optional - pycryptosat, long time sage: sls_aes = solve_sat(eqs, n = infinity) # optional - pycryptosat, long time sage: len(sls_aes) # optional - pycryptosat, long time 256
Note
Although supported, passing converter and solver objects instead of classes is discouraged because these objects are stateful.