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sbox.py
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sbox.py
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r"""
S-Boxes and Their Algebraic Representations
"""
from __future__ import print_function, division
from six.moves import range
from six import integer_types
from sage.combinat.integer_vector import IntegerVectors
from sage.crypto.boolean_function import BooleanFunction
from sage.matrix.constructor import Matrix
from sage.misc.cachefunc import cached_method
from sage.misc.functional import is_even
from sage.misc.misc_c import prod as mul
from sage.modules.free_module_element import vector
from sage.rings.finite_rings.element_base import is_FiniteFieldElement
from sage.rings.finite_rings.finite_field_constructor import FiniteField as GF
from sage.rings.ideal import FieldIdeal, Ideal
from sage.rings.integer_ring import ZZ
from sage.rings.integer import Integer
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.structure.sage_object import SageObject
class SBox(SageObject):
r"""
A substitution box or S-box is one of the basic components of
symmetric key cryptography. In general, an S-box takes ``m`` input
bits and transforms them into ``n`` output bits. This is called an
``mxn`` S-box and is often implemented as a lookup table. These
S-boxes are carefully chosen to resist linear and differential
cryptanalysis [He2002]_.
This module implements an S-box class which allows an algebraic
treatment and determine various cryptographic properties.
EXAMPLES:
We consider the S-box of the block cipher PRESENT [PRESENT07]_::
sage: S = mq.SBox(12,5,6,11,9,0,10,13,3,14,15,8,4,7,1,2); S
(12, 5, 6, 11, 9, 0, 10, 13, 3, 14, 15, 8, 4, 7, 1, 2)
sage: S(1)
5
Note that by default bits are interpreted in big endian
order. This is not consistent with the rest of Sage, which has a
strong bias towards little endian, but is consistent with most
cryptographic literature::
sage: S([0,0,0,1])
[0, 1, 0, 1]
sage: S = mq.SBox(12,5,6,11,9,0,10,13,3,14,15,8,4,7,1,2, big_endian=False)
sage: S(1)
5
sage: S([0,0,0,1])
[1, 1, 0, 0]
Now we construct an ``SBox`` object for the 4-bit small scale AES
S-Box (cf. :mod:`sage.crypto.mq.sr`)::
sage: sr = mq.SR(1,1,1,4, allow_zero_inversions=True)
sage: S = mq.SBox([sr.sub_byte(e) for e in list(sr.k)])
sage: S
(6, 5, 2, 9, 4, 7, 3, 12, 14, 15, 10, 0, 8, 1, 13, 11)
AUTHORS:
- Rusydi H. Makarim (2016-03-31) : added more functions to determine related cryptographic properties
- Yann Laigle-Chapuy (2009-07-01): improve linear and difference matrix computation
- Martin R. Albrecht (2008-03-12): initial implementation
REFERENCES:
- [He2002]_
- [PRESENT07]_
- [CDL2015]_
"""
def __init__(self, *args, **kwargs):
"""
Construct a substitution box (S-box) for a given lookup table
`S`.
INPUT:
- ``S`` - a finite iterable defining the S-box with integer or
finite field elements
- ``big_endian`` - controls whether bits shall be ordered in
big endian order (default: ``True``)
EXAMPLES:
We construct a 3-bit S-box where e.g. the bits (0,0,1) are
mapped to (1,1,1).::
sage: S = mq.SBox(7,6,0,4,2,5,1,3); S
(7, 6, 0, 4, 2, 5, 1, 3)
sage: S(0)
7
TESTS::
sage: S = mq.SBox()
Traceback (most recent call last):
...
TypeError: No lookup table provided.
sage: S = mq.SBox(1, 2, 3)
Traceback (most recent call last):
...
TypeError: Lookup table length is not a power of 2.
sage: S = mq.SBox(5, 6, 0, 3, 4, 2, 1, 2)
sage: S.n
3
"""
if "S" in kwargs:
S = kwargs["S"]
elif len(args) == 1:
S = args[0]
elif len(args) > 1:
S = args
else:
raise TypeError("No lookup table provided.")
_S = []
for e in S:
if is_FiniteFieldElement(e):
e = e.polynomial().change_ring(ZZ).subs( e.parent().characteristic() )
_S.append(e)
S = _S
if not ZZ(len(S)).is_power_of(2):
raise TypeError("Lookup table length is not a power of 2.")
self._S = S
self.m = ZZ(len(S)).exact_log(2)
self.n = ZZ(max(S)).nbits()
self._F = GF(2)
self._big_endian = kwargs.get("big_endian",True)
self.differential_uniformity = self.maximal_difference_probability_absolute
def _repr_(self):
"""
EXAMPLES::
sage: mq.SBox(7,6,0,4,2,5,1,3) #indirect doctest
(7, 6, 0, 4, 2, 5, 1, 3)
"""
return "(" + ", ".join(map(str,list(self))) + ")"
def __len__(self):
"""
Return the length of input bit strings.
EXAMPLES::
sage: len(mq.SBox(7,6,0,4,2,5,1,3))
3
"""
return self.m
def __eq__(self, other):
"""
S-boxes are considered to be equal if all construction
parameters match.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: loads(dumps(S)) == S
True
"""
return (self._S, self._big_endian) == (other._S, self._big_endian)
def __ne__(self, other):
"""
S-boxes are considered to be equal if all construction
parameters match.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S != S
False
"""
return not self.__eq__(other)
def to_bits(self, x, n=None):
"""
Return bitstring of length ``n`` for integer ``x``. The
returned bitstring is guaranteed to have length ``n``.
INPUT:
- ``x`` - an integer
- ``n`` - bit length (optional)
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.to_bits(6)
[1, 1, 0]
sage: S.to_bits( S(6) )
[0, 0, 1]
sage: S( S.to_bits( 6 ) )
[0, 0, 1]
"""
if n is None and self.m == self.n:
n = self.n
if self._big_endian:
swp = lambda x: list(reversed(x))
else:
swp = lambda x: x
return swp(self._rpad([self._F(_) for _ in ZZ(x).digits(2)], n))
def from_bits(self, x, n=None):
"""
Return integer for bitstring ``x`` of length ``n``.
INPUT:
- ``x`` - a bitstring
- ``n`` - bit length (optional)
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.from_bits( [1,1,0])
6
sage: S( S.from_bits( [1,1,0] ) )
1
sage: S.from_bits( S( [1,1,0] ) )
1
"""
if n is None and self.m == self.n:
n = self.m
if self._big_endian:
swp = lambda x: list(reversed(x))
else:
swp = lambda x: x
return ZZ( [ZZ(_) for _ in self._rpad(swp(x), n)], 2)
def _rpad(self,x, n=None):
"""
Right pads ``x`` such that ``len(x) == n``.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S._rpad([1,1])
[1, 1, 0]
"""
if n is None and self.m == self.n:
n = self.n
return x + [self._F(0)]*(n-len(x))
def __call__(self, X):
"""
Apply substitution to ``X``.
If ``X`` is a list, it is interpreted as a sequence of bits
depending on the bit order of this S-box.
INPUT:
- ``X`` - either an integer, a tuple of `\GF{2}` elements of
length ``len(self)`` or a finite field element in
`\GF{2^n}`. As a last resort this function tries to convert
``X`` to an integer.
EXAMPLES::
sage: S = mq.SBox([7,6,0,4,2,5,1,3])
sage: S(7)
3
sage: S((0,2,3))
[0, 1, 1]
sage: S[0]
7
sage: S[(0,0,1)]
[1, 1, 0]
sage: k.<a> = GF(2^3)
sage: S(a^2)
a
sage: S(QQ(3))
4
sage: S([1]*10^6)
Traceback (most recent call last):
...
TypeError: Cannot apply SBox to provided element.
sage: S(1/2)
Traceback (most recent call last):
...
TypeError: Cannot apply SBox to 1/2.
sage: S = mq.SBox(3, 0, 1, 3, 1, 0, 2, 2)
sage: S(0)
3
sage: S([0,0,0])
[1, 1]
"""
if isinstance(X, integer_types + (Integer,)):
return self._S[ZZ(X)]
try:
from sage.modules.free_module_element import vector
K = X.parent()
if K.order() == 2**self.n:
X = vector(X)
else:
raise TypeError
if not self._big_endian:
X = list(reversed(X))
else:
X = list(X)
X = ZZ([ZZ(_) for _ in X], 2)
out = self.to_bits(self._S[X], self.n)
if self._big_endian:
out = list(reversed(out))
return K(vector(GF(2),out))
except (AttributeError, TypeError):
pass
try:
if len(X) == self.m:
if self._big_endian:
X = list(reversed(X))
X = ZZ([ZZ(_) for _ in X], 2)
out = self._S[X]
return self.to_bits(out,self.n)
except TypeError:
pass
try:
return self._S[ZZ(X)]
except TypeError:
pass
if len(str(X)) > 50:
raise TypeError("Cannot apply SBox to provided element.")
else:
raise TypeError("Cannot apply SBox to %s."%(X,))
def __getitem__(self, X):
"""
See :meth:`SBox.__call__`.
EXAMPLES::
sage: S = mq.SBox([7,6,0,4,2,5,1,3])
sage: S[7]
3
"""
return self(X)
def is_permutation(self):
r"""
Return ``True`` if this S-Box is a permutation.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.is_permutation()
True
sage: S = mq.SBox(3,2,0,0,2,1,1,3)
sage: S.is_permutation()
False
"""
if self.m != self.n:
return False
return len(set([self(i) for i in range(2**self.m)])) == 2**self.m
def __iter__(self):
"""
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: [e for e in S]
[7, 6, 0, 4, 2, 5, 1, 3]
"""
for i in range(2**self.m):
yield self(i)
def difference_distribution_matrix(self):
"""
Return difference distribution matrix ``A`` for this S-box.
The rows of ``A`` encode the differences ``Delta I`` of the
input and the columns encode the difference ``Delta O`` for
the output. The bits are ordered according to the endianess of
this S-box. The value at ``A[Delta I,Delta O]`` encodes how
often ``Delta O`` is the actual output difference given
``Delta I`` as input difference.
See [He2002]_ for an introduction to differential
cryptanalysis.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.difference_distribution_matrix()
[8 0 0 0 0 0 0 0]
[0 2 2 0 2 0 0 2]
[0 0 2 2 0 0 2 2]
[0 2 0 2 2 0 2 0]
[0 2 0 2 0 2 0 2]
[0 0 2 2 2 2 0 0]
[0 2 2 0 0 2 2 0]
[0 0 0 0 2 2 2 2]
"""
m = self.m
n = self.n
nrows = 1<<m
ncols = 1<<n
A = Matrix(ZZ, nrows, ncols)
for i in range(nrows):
si = self(i)
for di in range(nrows):
A[ di , si^self(i^di)] += 1
return A
def maximal_difference_probability_absolute(self):
"""
Return the difference probability of the difference with the
highest probability in absolute terms, i.e. how often it
occurs in total.
Equivalently, this is equal to the differential uniformity
of this S-Box.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.maximal_difference_probability_absolute()
2
.. note::
This code is mainly called internally.
"""
A = self.difference_distribution_matrix().__copy__()
A[0,0] = 0
return max(map(abs, A.list()))
def maximal_difference_probability(self):
r"""
Return the difference probability of the difference with the
highest probability in the range between 0.0 and 1.0
indicating 0\% or 100\% respectively.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.maximal_difference_probability()
0.25
"""
return self.maximal_difference_probability_absolute()/(2.0**self.n)
@cached_method
def linear_approximation_matrix(self):
"""
Return linear approximation matrix ``A`` for this S-box.
Let ``i_b`` be the ``b``-th bit of ``i`` and ``o_b`` the
``b``-th bit of ``o``. Then ``v = A[i,o]`` encodes the bias of
the equation ``sum( i_b * x_i ) = sum( o_b * y_i )`` if
``x_i`` and ``y_i`` represent the input and output variables
of the S-box.
See [He2002]_ for an introduction to linear cryptanalysis.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.linear_approximation_matrix()
[ 4 0 0 0 0 0 0 0]
[ 0 0 0 0 2 2 2 -2]
[ 0 0 -2 -2 -2 2 0 0]
[ 0 0 -2 2 0 0 -2 -2]
[ 0 2 0 2 -2 0 2 0]
[ 0 -2 0 2 0 2 0 2]
[ 0 -2 -2 0 0 -2 2 0]
[ 0 -2 2 0 -2 0 0 -2]
According to this matrix the first bit of the input is equal
to the third bit of the output 6 out of 8 times::
sage: for i in srange(8): print(S.to_bits(i)[0] == S.to_bits(S(i))[2])
False
True
True
True
False
True
True
True
"""
m = self.m
n = self.n
nrows = 1<<m
ncols = 1<<n
B = BooleanFunction(self.m)
L = []
for j in range(ncols):
for i in range(nrows):
B[i] = ZZ(self(i)&j).popcount()
L.append(B.walsh_hadamard_transform())
A = Matrix(ZZ, ncols, nrows, L)
A = -A.transpose()/2
A.set_immutable()
return A
def maximal_linear_bias_absolute(self):
"""
Return maximal linear bias, i.e. how often the linear
approximation with the highest bias is true or false minus
`2^{n-1}`.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.maximal_linear_bias_absolute()
2
"""
A = self.linear_approximation_matrix().__copy__()
A[0,0] = 0
return max(map(abs, A.list()))
def maximal_linear_bias_relative(self):
"""
Return maximal bias of all linear approximations of this
S-box.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.maximal_linear_bias_relative()
0.25
"""
return self.maximal_linear_bias_absolute()/(2.0**self.m)
def ring(self):
"""
Create, return and cache a polynomial ring for S-box
polynomials.
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: S.ring()
Multivariate Polynomial Ring in x0, x1, x2, y0, y1, y2 over Finite Field of size 2
"""
try:
return self._ring
except AttributeError:
pass
m = self.m
n = self.n
X = range(m)
Y = range(n)
self._ring = PolynomialRing(self._F, m+n, ["x%d"%i for i in X] + ["y%d"%i for i in Y])
return self._ring
def solutions(self, X=None, Y=None):
"""
Return a dictionary of solutions to this S-box.
INPUT:
- ``X`` - input variables (default: ``None``)
- ``Y`` - output variables (default: ``None``)
EXAMPLES::
sage: S = mq.SBox([7,6,0,4,2,5,1,3])
sage: F = S.polynomials()
sage: s = S.solutions()
sage: any(f.subs(_s) for f in F for _s in s)
False
"""
if X is None and Y is None:
P = self.ring()
gens = P.gens()
else:
P = X[0].parent()
gens = X + Y
m = self.m
n = self.n
solutions = []
for i in range(1<<m):
solution = self.to_bits(i,m) + self( self.to_bits(i,m) )
solutions.append( dict(zip(gens, solution)) )
return solutions
def polynomials(self, X=None, Y=None, degree=2, groebner=False):
"""
Return a list of polynomials satisfying this S-box.
First, a simple linear fitting is performed for the given
``degree`` (cf. for example [BC2003]_). If ``groebner=True`` a
Groebner basis is also computed for the result of that
process.
INPUT:
- ``X`` - input variables
- ``Y`` - output variables
- ``degree`` - integer > 0 (default: ``2``)
- ``groebner`` - calculate a reduced Groebner basis of the
spanning polynomials to obtain more polynomials (default:
``False``)
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: P = S.ring()
By default, this method returns an indirect representation::
sage: S.polynomials()
[x0*x2 + x1 + y1 + 1,
x0*x1 + x1 + x2 + y0 + y1 + y2 + 1,
x0*y1 + x0 + x2 + y0 + y2,
x0*y0 + x0*y2 + x1 + x2 + y0 + y1 + y2 + 1,
x1*x2 + x0 + x1 + x2 + y2 + 1,
x0*y0 + x1*y0 + x0 + x2 + y1 + y2,
x0*y0 + x1*y1 + x1 + y1 + 1,
x1*y2 + x1 + x2 + y0 + y1 + y2 + 1,
x0*y0 + x2*y0 + x1 + x2 + y1 + 1,
x2*y1 + x0 + y1 + y2,
x2*y2 + x1 + y1 + 1,
y0*y1 + x0 + x2 + y0 + y1 + y2,
y0*y2 + x1 + x2 + y0 + y1 + 1,
y1*y2 + x2 + y0]
We can get a direct representation by computing a
lexicographical Groebner basis with respect to the right
variable ordering, i.e. a variable ordering where the output
bits are greater than the input bits::
sage: P.<y0,y1,y2,x0,x1,x2> = PolynomialRing(GF(2),6,order='lex')
sage: S.polynomials([x0,x1,x2],[y0,y1,y2], groebner=True)
[y0 + x0*x1 + x0*x2 + x0 + x1*x2 + x1 + 1,
y1 + x0*x2 + x1 + 1,
y2 + x0 + x1*x2 + x1 + x2 + 1]
"""
def nterms(nvars, deg):
"""
Return the number of monomials possible up to a given
degree.
INPUT:
- ``nvars`` - number of variables
- ``deg`` - degree
TESTS::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: F = S.polynomials(degree=3) # indirect doctest
"""
total = 1
divisor = 1
var_choices = 1
for d in range(1, deg+1):
var_choices *= (nvars - d + 1)
divisor *= d
total += var_choices/divisor
return total
m = self.m
n = self.n
F = self._F
if X is None and Y is None:
P = self.ring()
X = P.gens()[:m]
Y = P.gens()[m:]
else:
P = X[0].parent()
gens = X+Y
bits = []
for i in range(1<<m):
bits.append( self.to_bits(i,m) + self(self.to_bits(i,m)) )
ncols = (1<<m)+1
A = Matrix(P, nterms(m+n, degree), ncols)
exponents = []
for d in range(degree+1):
exponents += IntegerVectors(d, max_length=m+n, min_length=m+n, min_part=0, max_part=1).list()
row = 0
for exponent in exponents:
A[row,ncols-1] = mul([gens[i]**exponent[i] for i in range(len(exponent))])
for col in range(1<<m):
A[row,col] = mul([bits[col][i] for i in range(len(exponent)) if exponent[i]])
row +=1
for c in range(ncols):
A[0,c] = 1
RR = A.echelon_form(algorithm='row_reduction')
# extract spanning stet
gens = (RR.column(ncols-1)[1<<m:]).list()
if not groebner:
return gens
FI = set(FieldIdeal(P).gens())
I = Ideal(gens + list(FI))
gb = I.groebner_basis()
gens = []
for f in gb:
if f not in FI: # filter out field equations
gens.append(f)
return gens
def interpolation_polynomial(self, k=None):
r"""
Return a univariate polynomial over an extension field
representing this S-box.
If ``m`` is the input length of this S-box then the extension
field is of degree ``m``.
If the output length does not match the input length then a
``TypeError`` is raised.
INPUT:
- ``k`` - an instance of `\GF{2^m}` (default: ``None``)
EXAMPLES::
sage: S = mq.SBox(7,6,0,4,2,5,1,3)
sage: f = S.interpolation_polynomial()
sage: f
x^6 + a*x^5 + (a + 1)*x^4 + (a^2 + a + 1)*x^3
+ (a^2 + 1)*x^2 + (a + 1)*x + a^2 + a + 1
sage: a = f.base_ring().gen()
sage: f(0), S(0)
(a^2 + a + 1, 7)
sage: f(a^2 + 1), S(5)
(a^2 + 1, 5)
"""
if self.m != self.n:
raise TypeError("Lagrange interpolation only supported if self.m == self.n.")
if k is None:
k = GF(2**self.m,'a')
l = []
for i in range(2**self.m):
i = self.to_bits(i, self.m)
o = self(i)
if self._big_endian:
i = reversed(i)
o = reversed(o)
l.append( (k(vector(i)), k(vector(o))) )
P = PolynomialRing(k,'x')
return P.lagrange_polynomial(l)
def cnf(self, xi=None, yi=None, format=None):
"""
Return a representation of this S-Box in conjunctive normal
form.
This function examines the truth tables for each output bit of
the S-Box and thus has complexity `n * 2^m` for an ``m x n``
S-Box.
INPUT:
- ``xi`` - indices for the input variables (default: ``1...m``)
- ``yi`` - indices for the output variables (default: ``m+1 ... m+n``)
- ``format`` - output format, see below (default: ``None``)
FORMATS:
- ``None`` - return a list of tuples of integers where each
tuple represents a clause, the absolute value of an integer
represents a variable and the sign of an integer indicates
inversion.
- ``symbolic`` - a string that can be parsed by the
``SymbolicLogic`` package.
- ``dimacs`` - a string in DIMACS format which is the gold
standard for SAT-solver input (cf. http://www.satlib.org/).
- ``dimacs_headless`` - a string in DIMACS format, but without
the header. This is useful for concatenation of outputs.
EXAMPLES:
We give a very small example to explain the output format::
sage: S = mq.SBox(1,2,0,3); S
(1, 2, 0, 3)
sage: cnf = S.cnf(); cnf
[(1, 2, -3), (1, 2, 4),
(1, -2, 3), (1, -2, -4),
(-1, 2, -3), (-1, 2, -4),
(-1, -2, 3), (-1, -2, 4)]
This output completely describes the S-Box. For instance, we
can check that ``S([0,1]) -> [1,0]`` satisfies every clause if
the first input bit corresponds to the index ``1`` and the
last output bit corresponds to the index ``3`` in the
output.
We can convert this representation to the DIMACS format::
sage: print(S.cnf(format='dimacs'))
p cnf 4 8
1 2 -3 0
1 2 4 0
1 -2 3 0
1 -2 -4 0
-1 2 -3 0
-1 2 -4 0
-1 -2 3 0
-1 -2 4 0
For concatenation we can strip the header::
sage: print(S.cnf(format='dimacs_headless'))
1 2 -3 0
1 2 4 0
1 -2 3 0
1 -2 -4 0
-1 2 -3 0
-1 2 -4 0
-1 -2 3 0
-1 -2 4 0
This might be helpful in combination with the ``xi`` and
``yi`` parameter to assign indices manually::
sage: print(S.cnf(xi=[10,20],yi=[30,40], format='dimacs_headless'))
10 20 -30 0
10 20 40 0
10 -20 30 0
10 -20 -40 0
-10 20 -30 0
-10 20 -40 0
-10 -20 30 0
-10 -20 40 0
We can also return a string which is parse-able by the
``SymbolicLogic`` package::
sage: log = SymbolicLogic()
sage: s = log.statement(S.cnf(format='symbolic'))
sage: log.truthtable(s)[1:]
[['False', 'False', 'False', 'False', 'False'],
['False', 'False', 'False', 'True', 'False'],
['False', 'False', 'True', 'False', 'False'],
['False', 'False', 'True', 'True', 'True'],
['False', 'True', 'False', 'False', 'True'],
['False', 'True', 'False', 'True', 'True'],
['False', 'True', 'True', 'False', 'True'],
['False', 'True', 'True', 'True', 'True'],
['True', 'False', 'False', 'False', 'True'],
['True', 'False', 'False', 'True', 'True'],
['True', 'False', 'True', 'False', 'True'],
['True', 'False', 'True', 'True', 'True'],
['True', 'True', 'False', 'False', 'True'],
['True', 'True', 'False', 'True', 'True'],
['True', 'True', 'True', 'False', 'True'],
['True', 'True', 'True', 'True', 'True']]
This function respects endianness of the S-Box::
sage: S = mq.SBox(1,2,0,3, big_endian=False); S
(1, 2, 0, 3)
sage: cnf = S.cnf(); cnf
[(1, 2, -4), (1, 2, 3),
(-1, 2, 4), (-1, 2, -3),
(1, -2, -4), (1, -2, -3),
(-1, -2, 4), (-1, -2, 3)]
S-Boxes with m!=n also work:
sage: o = list(range(8)) + list(range(8))
sage: shuffle(o)
sage: S = mq.SBox(o)
sage: S.is_permutation()
False
sage: len(S.cnf()) == 3*2^4
True
TESTS:
sage: S = mq.SBox(1,2,0,3, big_endian=False)
sage: S.cnf([1000,1001,1002], [2000,2001,2002])
Traceback (most recent call last):
...
TypeError: first arg required to have length 2, got 3 instead.
"""
m, n = self.m, self.n
if xi is None:
xi = [i+1 for i in range(m)]
if yi is None:
yi = [m+i+1 for i in range(n)]
if len(xi) != m:
raise TypeError("first arg required to have length %d, got %d instead."%(m,len(xi)))
if len(yi) != n:
raise TypeError("second arg required to have length %d, got %d instead."%(n,len(yi)))
output_bits = range(n)
if not self._big_endian:
output_bits = list(reversed(output_bits))
C = [] # the set of clauses
for e in range(2**m):
x = self.to_bits(e, m)
y = self(x) # evaluate at x
for output_bit in output_bits: # consider each bit
clause = [(-1)**(int(v)) * i for v,i in zip(x, xi)]
clause.append( (-1)**(1-int(y[output_bit])) * yi[output_bit] )
C.append(tuple(clause))
if format is None:
return C
elif format == 'symbolic':
gd = self.ring().gens()
formula = []
for clause in C:
clause = "|".join([str(gd[abs(v)-1]).replace("-","~") for v in clause])
formula.append("("+clause+")")
return " & ".join(formula)
elif format.startswith('dimacs'):
if format == "dimacs_headless":
header = ""
else:
header = "p cnf %d %d\n"%(m+n,len(C))
values = " 0\n".join([" ".join(map(str,line)) for line in C])
return header + values + " 0\n"
else:
raise ValueError("Format '%s' not supported."%(format,))