# 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 `solver` on SAT-instance produced by `converter` from `F`.

INPUT:

• `F` - a sequence of Boolean polynomials

• `converter` - an ANF to CNF converter class or object. If `converter` is `None` then `sage.sat.converters.polybori.CNFEncoder` is used to construct a new converter. (default: `None`)

• `solver` - a SAT-solver class or object. If `solver` is `None` then `sage.sat.solvers.cryptominisat.CryptoMiniSat` is used to construct a new converter. (default: `None`)

• `max_learnt_length` - only clauses of length <= `max_length_learnt` are 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_` and `s_` respectively. For example, to increase CryptoMiniSat’s verbosity level, pass `s_verbosity=1`.

OUTPUT:

A sequence of Boolean polynomials.

EXAMPLES:

```sage: from sage.sat.boolean_polynomials import learn as learn_sat # optional - cryptominisat
```

We construct a simple system and solve it:

```sage: set_random_seed(2300)                      # optional - cryptominisat
sage: sr = mq.SR(1,2,2,4,gf2=True,polybori=True) # optional - cryptominisat
sage: F,s = sr.polynomial_system()               # optional - cryptominisat
sage: H = learn_sat(F)                           # optional - cryptominisat
sage: H[-1]                                      # optional - cryptominisat
k033 + 1
```
sage.sat.boolean_polynomials.solve(F, converter=None, solver=None, n=1, target_variables=None, **kwds)

Solve system of Boolean polynomials `F` by solving the SAT-problem – produced by `converter` – using `solver`.

INPUT:

• `F` - a sequence of Boolean polynomials

• `n` - number of solutions to return. If `n` is +infinity then all solutions are returned. If `n <infinity` then `n` solutions are returned if `F` has at least `n` solutions. Otherwise, all solutions of `F` are returned. (default: `1`)

• `converter` - an ANF to CNF converter class or object. If `converter` is `None` then `sage.sat.converters.polybori.CNFEncoder` is used to construct a new converter. (default: `None`)

• `solver` - a SAT-solver class or object. If `solver` is `None` then `sage.sat.solvers.cryptominisat.CryptoMiniSat` is 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. Furthermore `target_variables` denotes which variable-value pairs appear in the solutions. If `target_variables` is `None` all variables appearing in the polynomials of `F` are used to construct exclusion clauses. (default: `None`)

• `**kwds` - parameters can be passed to the converter and the

solver by prefixing them with `c_` and `s_` respectively. For example, to increase CryptoMiniSat’s verbosity level, pass `s_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: F,s = sr.polynomial_system()
```

and pass it to a SAT solver:

```sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - cryptominisat
sage: s = solve_sat(F)                                            # optional - cryptominisat
sage: F.subs(s)                                                # optional - cryptominisat
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, s_verbosity=1, c_max_vars_sparse=4, c_cutting_number=8) # optional - cryptominisat
c ...
...
sage: F.subs(s)                                                             # optional - cryptominisat
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 - cryptominisat
sage: f = a*b                           # optional - cryptominisat
sage: l = solve_sat([f],n=1)            # optional - cryptominisat
sage: len(l) == 1, f.subs(l)         # optional - cryptominisat
(True, 0)

sage: l = solve_sat([a*b],n=2)        # optional - cryptominisat
sage: len(l) == 2, f.subs(l), f.subs(l) # optional - cryptominisat
(True, 0, 0)

sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=3))  # optional - cryptominisat
[(0, 0), (0, 1), (1, 0)]
sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=4))   # optional - cryptominisat
[(0, 0), (0, 1), (1, 0)]
sage: sorted((d[a], d[b]) for d in solve_sat([a*b],n=infinity))  # optional - cryptominisat
[(0, 0), (0, 1), (1, 0)]
```

In the next example we see how the `target_variables` parameter works:

```sage: from sage.sat.boolean_polynomials import solve as solve_sat # optional - cryptominisat
sage: R.<a,b,c,d> = BooleanPolynomialRing()                       # optional - cryptominisat
sage: F = [a+b,a+c+d]                                             # optional - cryptominisat
```

First the normal use case:

```sage: sorted((D[a], D[b], D[c], D[d]) for D in solve_sat(F,n=infinity))      # optional - cryptominisat
[(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 - cryptominisat
[{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 - cryptominisat, long time
sage: from sage.sat.boolean_polynomials import solve as solve_sat                       # optional - cryptominisat, long time
sage: sr = sage.crypto.mq.SR(1, 4, 4, 8, allow_zero_inversions = True)                  # optional - cryptominisat, long time
sage: sb = sr.sbox()                                                                    # optional - cryptominisat, long time
sage: eqs = sb.polynomials(degree = 3)                                                  # optional - cryptominisat, long time
sage: eqs = PolynomialSequence(eqs)                                                     # optional - cryptominisat, long time
sage: variables = map(str, eqs.variables())                                             # optional - cryptominisat, long time
sage: variables = ",".join(variables)                                                   # optional - cryptominisat, long time
sage: R = BooleanPolynomialRing(16, variables)                                          # optional - cryptominisat, long time
sage: eqs = [R(eq) for eq in eqs]                                                                 # optional - cryptominisat, long time
sage: sls_aes = solve_sat(eqs, n = infinity)                                            # optional - cryptominisat, long time
sage: len(sls_aes)                                                                      # optional - cryptominisat, long time
256
```

Note

Although supported, passing converter and solver objects instead of classes is discouraged because these objects are stateful.