r"""
Small Scale Variants of the AES (SR) Polynomial System Generator
Sage supports polynomial system generation for small scale (and full
scale) AES variants over `\GF{2}` and `\GF{2^e}`. Also, Sage supports
both the specification of SR as given in the papers [CMR05]_ and
[CMR06]_ and a variant of SR* which is equivalent to AES.
SR is a family of parameterizable variants of the AES suitable as a
framework for comparing different cryptanalytic techniques that can be
brought to bear on the AES. It is different from
:class:`Mini-AES <sage.crypto.block_cipher.miniaes.MiniAES>`, whose
purpose is as a teaching tool to help beginners understand the basic
structure and working of the full AES.
AUTHORS:
- Martin Albrecht (2008,2009-01): usability improvements
- Martin Albrecht (2007-09): initial version
- Niles Johnson (2010-08): Trac #3893: ``random_element()`` should pass on ``*args`` and ``**kwds``.
EXAMPLES:
We construct SR(1,1,1,4) and study its properties.
::
sage: sr = mq.SR(1, 1, 1, 4)
``n`` is the number of rounds, ``r`` the number of rows in the
state array, ``c`` the number of columns in the state array, and ``e`` the
degree of the underlying field.
::
sage: sr.n, sr.r, sr.c, sr.e
(1, 1, 1, 4)
By default variables are ordered reverse to as they appear, e.g.::
sage: print sr.R.repr_long()
Polynomial Ring
Base Ring : Finite Field in a of size 2^4
Size : 20 Variables
Block 0 : Ordering : deglex
Names : k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, w103, s000, s001, s002, s003, k000, k001, k002, k003
However, this can be prevented by passing in ``reverse_variables=False`` to the constructor.
For SR(1, 1, 1, 4) the ``ShiftRows`` matrix isn't that interesting.::
sage: sr.ShiftRows
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
Also, the ``MixColumns`` matrix is the identity matrix.::
sage: sr.MixColumns
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
``Lin``, however, is not the identity matrix.::
sage: sr.Lin
[ a^2 + 1 1 a^3 + a^2 a^2 + 1]
[ a a 1 a^3 + a^2 + a + 1]
[ a^3 + a a^2 a^2 1]
[ 1 a^3 a + 1 a + 1]
``M`` and ``Mstar`` are identical for SR(1, 1, 1, 4)::
sage: sr.M
[ a^2 + 1 1 a^3 + a^2 a^2 + 1]
[ a a 1 a^3 + a^2 + a + 1]
[ a^3 + a a^2 a^2 1]
[ 1 a^3 a + 1 a + 1]
::
sage: sr.Mstar
[ a^2 + 1 1 a^3 + a^2 a^2 + 1]
[ a a 1 a^3 + a^2 + a + 1]
[ a^3 + a a^2 a^2 1]
[ 1 a^3 a + 1 a + 1]
However, for larger instances of SR Mstar is not equal to M::
sage: sr = mq.SR(10,4,4,8)
sage: sr.Mstar == ~sr.MixColumns * sr.M
True
We can compute a Groebner basis for the ideals spanned by SR
instances to recover all solutions to the system.::
sage: sr = mq.SR(1,1,1,4, gf2=True, polybori=True)
sage: K = sr.base_ring()
sage: a = K.gen()
sage: K = [a]
sage: P = [1]
sage: F,s = sr.polynomial_system(P=P, K=K)
sage: F.groebner_basis()
[k100, k101 + 1, k102, k103 + k003,
x100 + 1, x101 + k003 + 1, x102 + k003 + 1,
x103 + k003, w100, w101, w102 + 1, w103 + k003 + 1,
s000 + 1, s001 + k003, s002 + k003, s003 + k003 + 1,
k000, k001, k002 + 1]
Note that the order of ``k000``, ``k001``, ``k002`` and ``k003`` is
little endian. Thus the result ``k002 + 1, k001, k000`` indicates that
the key is either `a` or `a+1`. We can verify that both keys encrypt P
to the same ciphertext::
sage: sr(P,[a])
[0]
sage: sr(P,[a+1])
[0]
All solutions can easily be recovered using the variety function for ideals.::
sage: I = F.ideal()
sage: for V in I.variety():
... for k,v in sorted(V.iteritems()):
... print k,v
... print
k003 0
k002 1
k001 0
k000 0
s003 1
s002 0
s001 0
s000 1
w103 1
w102 1
w101 0
w100 0
x103 0
x102 1
x101 1
x100 1
k103 0
k102 0
k101 1
k100 0
<BLANKLINE>
k003 1
k002 1
k001 0
k000 0
s003 0
s002 1
s001 1
s000 1
w103 0
w102 1
w101 0
w100 0
x103 1
x102 0
x101 0
x100 1
k103 1
k102 0
k101 1
k100 0
We can also verify the correctness of the variety by evaluating all
ideal generators on all points.::
sage: for V in I.variety():
... for f in I.gens():
... if f.subs(V) != 0:
... print "epic fail"
Note that the S-Box object for SR can be constructed with a call to ``sr.sbox()``::
sage: sr = mq.SR(1,1,1,4, gf2=True, polybori=True)
sage: S = sr.sbox()
For example, we can now study the difference distribution matrix of ``S``::
sage: S.difference_distribution_matrix()
[16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[ 0 2 2 2 2 0 0 0 2 0 0 0 2 4 0 0]
[ 0 2 0 4 2 2 2 0 0 2 0 0 0 0 0 2]
[ 0 2 4 0 0 2 0 0 2 2 0 2 0 0 2 0]
[ 0 0 2 0 4 2 0 0 0 0 2 0 2 0 2 2]
[ 0 0 0 2 0 0 0 2 4 2 0 0 2 0 2 2]
[ 0 4 0 0 0 2 0 2 0 2 2 0 2 2 0 0]
[ 0 2 0 0 0 0 2 0 0 0 0 2 4 2 2 2]
[ 0 2 2 0 0 0 2 2 2 0 2 0 0 0 0 4]
[ 0 0 2 2 0 0 0 0 0 2 2 4 0 2 0 2]
[ 0 0 2 0 2 0 2 2 0 4 0 2 2 0 0 0]
[ 0 0 0 0 2 0 2 0 2 2 4 0 0 2 2 0]
[ 0 0 0 2 0 4 2 0 2 0 2 2 2 0 0 0]
[ 0 0 0 0 2 2 0 4 2 0 0 2 0 2 0 2]
[ 0 0 2 2 0 2 4 2 0 0 0 0 0 2 2 0]
[ 0 2 0 2 2 0 0 2 0 0 2 2 0 0 4 0]
or use ``S`` to find alternative polynomial representations for the S-Box.::
sage: S.polynomials(degree=3)
[x0*x1 + x1*x2 + x0*x3 + x0*y2 + x1 + y0 + y1 + 1,
x0*x1 + x0*x2 + x0*y0 + x0*y1 + x0*y2 + x1 + x2 + y0 + y1 + y2,
x0*x1 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x1*y0 + x0*y1 + x0*y3,
x0*x1 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x1*y1 + x0*y3 + x1 + y0 + y1 + 1,
x0*x1 + x0*x2 + x0*y2 + x1*y2 + x0*y3 + x0 + x1,
x0*x3 + x1*x3 + x0*y1 + x0*y2 + x1*y3 + x0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x0*x1 + x1*x3 + x2*x3 + x0*y0 + x0*y2 + x0*y3 + x2 + y0 + y3,
x0*x1 + x0*x2 + x0*x3 + x1*x3 + x2*y0 + x0*y2 + x0 + x2 + x3 + y3,
x0*x3 + x1*x3 + x0*y0 + x2*y1 + x0*y2 + x3 + y3,
x0*x1 + x0*x2 + x0*y0 + x0*y1 + x2*y2 + x0*y3 + x1 + y0 + y1 + 1,
x0*x3 + x1*x3 + x0*y0 + x0*y1 + x0*y3 + x2*y3 + y0 + y3,
x0*x1 + x0*x2 + x3*y0 + x0*y1 + x0*y3 + y0,
x0*y0 + x0*y1 + x3*y1 + x0 + x2 + y0 + y3,
x0*y0 + x3*y2 + y0,
x0*x1 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x0*y2 + x3*y3 + y0,
x0*x2 + x0*x3 + x0*y1 + y0*y1 + x0*y3 + x2 + x3 + y3,
x0*x2 + x0*y0 + y0*y2 + x0*y3 + x0 + y0,
x0*x1 + x0*x2 + x1*x3 + x0*y2 + y0*y3 + y0,
x0*x1 + x0*y0 + y1*y2 + x0*y3 + x1 + x2 + y0 + 1,
x0*x2 + x1*x3 + x0*y1 + x0*y2 + x0*y3 + y1*y3 + x0 + y0 + y3,
x0*x1 + x0*x2 + x0*x3 + x0*y1 + x0*y2 + x0*y3 + y2*y3 + x0 + x1 + x2 + x3 + y1 + y3 + 1,
x0*x1*x2 + x0*x3 + x0*y0 + x0*y1 + x0*y2 + x0,
x0*x1*x3 + x0*x2 + x0*x3 + x0*y1 + x0*y3 + x0,
x0*x1*y0 + x0*x1 + x0*y0 + x0,
x0*x1*y1,
x0*x1*y2 + x0*x2 + x0*y2 + x0*y3 + x0,
x0*x1*y3 + x0*x1 + x0*x3 + x0*y0 + x0*y1 + x0*y2 + x0,
x0*x2*x3 + x0*x1 + x0*x3 + x0*y1 + x0*y2 + x0*y3 + x0,
x0*x2*y0 + x0*x1 + x0*x2 + x0*x3 + x0*y1 + x0*y2,
x0*x2*y1 + x0*x2 + x0*x3 + x0*y0 + x0*y1 + x0*y2 + x0,
x0*x2*y2 + x0*x2 + x0*y3 + x0,
x0*x2*y3 + x0*x2 + x0*y3 + x0,
x0*x3*y0 + x0*x1 + x0*x2 + x0*y0 + x0*y1 + x0*y3,
x0*x3*y1 + x0*x2 + x0*y1 + x0*y3 + x0,
x0*x3*y2,
x0*x3*y3 + x0*x1 + x0*y1 + x0*y2 + x0*y3 + x0,
x0*y0*y1 + x0*y1,
x0*y0*y2 + x0*x2 + x0*y3 + x0,
x0*y0*y3 + x0*x1 + x0*x3 + x0*y0 + x0*y1 + x0*y2 + x0*y3 + x0,
x0*y1*y2 + x0*x2 + x0*y3 + x0,
x0*y1*y3 + x0*x3 + x0*y0 + x0*y2 + x0*y3,
x0*y2*y3 + x0*y2,
x1*x2*x3 + x0*x1 + x1*x3 + x0*y0 + x0*y1 + x2 + x3 + y3,
x1*x2*y0 + x0*x1 + x1*x3 + x0*y0 + x0*y1 + x2 + x3 + y3,
x1*x2*y1 + x0*x1 + x1*x3 + x0*y0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x1*x2*y2 + x0*x1 + x0*y0 + x0*y1 + x0 + x1 + y0 + y1 + 1,
x1*x2*y3 + x0*x1 + x1*x3 + x0*y0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x1*x3*y0 + x0*x1 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x0*y1 + x0*y3,
x1*x3*y1 + x0*x2 + x0*x3 + x0*y3 + x2 + x3 + y3,
x1*x3*y2 + x0*x2 + x0*x3 + x1*x3 + x0*y1 + x0*y3 + x0,
x1*x3*y3 + x0*x1 + x0*x2 + x0*x3 + x0*y0 + x0*y1 + x0*y3,
x1*y0*y1 + x0*x2 + x0*x3 + x0*y3 + x2 + x3 + y3,
x1*y0*y2 + x0*x2 + x0*x3 + x1*x3 + x0*y1 + x0*y3 + x0,
x1*y0*y3,
x1*y1*y2 + x0*x1 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x0*y3 + x1 + y0 + y1 + 1,
x1*y1*y3 + x0*x1 + x1*x3 + x0*y0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x1*y2*y3 + x0*x1 + x0*x2 + x1*x3 + x0*y0 + x0*y2 + x0*y3 + x0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x2*x3*y0 + x0*x1 + x0*x3 + x1*x3 + x0*y2 + x0*y3 + x2 + x3 + y3,
x2*x3*y1 + x0*y1 + x0*y2 + x0*y3 + x3 + y0,
x2*x3*y2 + x1*x3 + x0*y1 + x0 + x2 + x3 + y3,
x2*x3*y3,
x2*y0*y1 + x0*x2 + x0*x3 + x0*y0 + x0*y1 + x0*y2 + x0,
x2*y0*y2 + x0*x2 + x1*x3 + x0*y1 + x0*y3 + x2 + x3 + y3,
x2*y0*y3 + x0*x2 + x0*y3 + x0,
x2*y1*y2 + x0*x1 + x0*x2 + x1*x3 + x0*y0 + x0*y3 + x0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x2*y1*y3 + x0*x3 + x1*x3 + x0*y0 + x0*y1 + x0*y3 + y0 + y3,
x2*y2*y3 + x0*x1 + x0*x2 + x1*x3 + x0*y0 + x0*y3 + x0 + x1 + x2 + x3 + y0 + y1 + y3 + 1,
x3*y0*y1 + x0*x3 + x0*y1 + x0 + x2 + x3 + y3,
x3*y0*y2 + x0*y0 + y0,
x3*y0*y3 + x1*x3 + x0*y1 + x0*y2 + x0*y3 + y0,
x3*y1*y2 + x0*x2 + x0*x3 + x0*y3 + x2 + x3 + y3,
x3*y1*y3 + x0*x2 + x0*x3 + x0*y0 + x0*y2 + x0,
x3*y2*y3 + x0*x2 + x0*x3 + x1*x3 + x0*y0 + x0*y1 + x0*y3 + x0 + y0,
y0*y1*y2 + x0*x3 + x0 + x2 + x3 + y3,
y0*y1*y3 + x0*x3 + x0*y0 + x0*y2 + x0*y3,
y0*y2*y3 + x0*x3 + x1*x3 + x0*y0 + x0*y1 + y0,
y1*y2*y3 + x0*x1 + x0*x2 + x1*x3 + x0*y0 + x0*y3 + x0 + x1 + x2 + x3 + y0 + y1 + y3 + 1]
sage: S.interpolation_polynomial()
(a^2 + 1)*x^14 + x^13 + (a^3 + a^2)*x^11 + (a^2 + 1)*x^7 + a^2 + a
The :class:`SR_gf2_2` gives an example how use alternative polynomial
representations of the S-Box for construction of polynomial systems.
TESTS::
sage: sr == loads(dumps(sr))
True
REFERENCES:
.. [CMR05] C\. Cid, S\. Murphy, M\. Robshaw *Small Scale Variants of
the AES*\; in Proceedings of Fast Software Encryption 2005\; LNCS
3557\; Springer Verlag 2005\; available at
http://www.isg.rhul.ac.uk/~sean/smallAES-fse05.pdf
.. [CMR06] C\. Cid, S\. Murphy, and M\. Robshaw *Algebraic Aspects of
the Advanced Encryption Standard*\; Springer Verlag 2006
.. [MR02] S\. Murphy, M\. Robshaw *Essential Algebraic Structure
Within the AES*\; in Advances in Cryptology \- CRYPTO 2002\; LNCS
2442\; Springer Verlag 2002
"""
from sage.rings.finite_rings.constructor import FiniteField as GF
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing, BooleanPolynomialRing_constructor as BooleanPolynomialRing
from sage.matrix.matrix import is_Matrix
from sage.matrix.constructor import Matrix, random_matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.misc import get_verbose
from sage.misc.flatten import flatten
from sage.modules.vector_modn_dense import Vector_modn_dense
from sage.rings.polynomial.multi_polynomial_sequence import PolynomialSequence
from mpolynomialsystemgenerator import MPolynomialSystemGenerator
from sage.rings.polynomial.term_order import TermOrder
def SR(n=1, r=1, c=1, e=4, star=False, **kwargs):
r"""
Return a small scale variant of the AES polynomial system
constructor subject to the following conditions:
INPUT:
- ``n`` - the number of rounds (default: 1)
- ``r`` - the number of rows in the state array (default: 1)
- ``c`` - the number of columns in the state array (default: 1)
- ``e`` - the exponent of the finite extension field (default: 4)
- ``star`` - determines if SR\* or SR should be constructed (default: ``False``)
- ``aes_mode`` - as the SR key schedule specification differs
slightly from the AES key schedule, this parameter controls
which schedule to use (default: ``True``)
- ``gf2`` - generate polynomial systems over `\GF{2}` rather than
over `\GF{2^e}` (default: ``False``)
- ``polybori`` - use the ``BooleanPolynomialRing`` as polynomial
representation (default: ``True``, `\GF{2}` only)
- ``order`` - a string to specify the term ordering of the
variables (default: ``deglex``)
- ``postfix`` - a string which is appended after the variable name
(default: '')
- ``allow_zero_inversions`` - a boolean to control whether zero
inversions raise an exception (default: ``False``)
- ``correct_only`` - only include correct inversion polynomials
(default: ``False``, `\GF{2}` only)
- ``biaffine_only`` - only include bilinear and biaffine inversion
polynomials (default: ``True``, `\GF{2}` only)
EXAMPLES::
sage: sr = mq.SR(1, 1, 1, 4)
sage: ShiftRows = sr.shift_rows_matrix()
sage: MixColumns = sr.mix_columns_matrix()
sage: Lin = sr.lin_matrix()
sage: M = MixColumns * ShiftRows * Lin
sage: print sr.hex_str_matrix(M)
5 1 C 5
2 2 1 F
A 4 4 1
1 8 3 3
::
sage: sr = mq.SR(1, 2, 1, 4)
sage: ShiftRows = sr.shift_rows_matrix()
sage: MixColumns = sr.mix_columns_matrix()
sage: Lin = sr.lin_matrix()
sage: M = MixColumns * ShiftRows * Lin
sage: print sr.hex_str_matrix(M)
F 3 7 F A 2 B A
A A 5 6 8 8 4 9
7 8 8 2 D C C 3
4 6 C C 5 E F F
A 2 B A F 3 7 F
8 8 4 9 A A 5 6
D C C 3 7 8 8 2
5 E F F 4 6 C C
::
sage: sr = mq.SR(1, 2, 2, 4)
sage: ShiftRows = sr.shift_rows_matrix()
sage: MixColumns = sr.mix_columns_matrix()
sage: Lin = sr.lin_matrix()
sage: M = MixColumns * ShiftRows * Lin
sage: print sr.hex_str_matrix(M)
F 3 7 F 0 0 0 0 0 0 0 0 A 2 B A
A A 5 6 0 0 0 0 0 0 0 0 8 8 4 9
7 8 8 2 0 0 0 0 0 0 0 0 D C C 3
4 6 C C 0 0 0 0 0 0 0 0 5 E F F
A 2 B A 0 0 0 0 0 0 0 0 F 3 7 F
8 8 4 9 0 0 0 0 0 0 0 0 A A 5 6
D C C 3 0 0 0 0 0 0 0 0 7 8 8 2
5 E F F 0 0 0 0 0 0 0 0 4 6 C C
0 0 0 0 A 2 B A F 3 7 F 0 0 0 0
0 0 0 0 8 8 4 9 A A 5 6 0 0 0 0
0 0 0 0 D C C 3 7 8 8 2 0 0 0 0
0 0 0 0 5 E F F 4 6 C C 0 0 0 0
0 0 0 0 F 3 7 F A 2 B A 0 0 0 0
0 0 0 0 A A 5 6 8 8 4 9 0 0 0 0
0 0 0 0 7 8 8 2 D C C 3 0 0 0 0
0 0 0 0 4 6 C C 5 E F F 0 0 0 0
"""
if not kwargs.get("gf2", False):
return SR_gf2n(n, r, c, e, star, **kwargs)
else:
return SR_gf2(n, r, c, e, star, **kwargs)
class SR_generic(MPolynomialSystemGenerator):
def __init__(self, n=1, r=1, c=1, e=4, star=False, **kwargs):
"""
Small Scale Variants of the AES.
EXAMPLES::
sage: sr = mq.SR(1, 1, 1, 4)
sage: ShiftRows = sr.shift_rows_matrix()
sage: MixColumns = sr.mix_columns_matrix()
sage: Lin = sr.lin_matrix()
sage: M = MixColumns * ShiftRows * Lin
sage: print sr.hex_str_matrix(M)
5 1 C 5
2 2 1 F
A 4 4 1
1 8 3 3
"""
if n-1 not in range(10):
raise TypeError, "n must be between 1 and 10 (inclusive)"
self._n = n
if r not in (1, 2, 4):
raise TypeError, "r must be in (1, 2, 4)"
self._r = r
if c not in (1, 2, 4):
raise TypeError, "c must be in (1, 2, 4)"
self._c = c
if e not in (4, 8):
raise TypeError, "e must be either 4 or 8"
self._e = e
self._star = bool(star)
self._base = self.base_ring()
self._postfix = kwargs.get("postfix", "")
self._order = kwargs.get("order", "deglex")
self._aes_mode = kwargs.get("aes_mode", True)
self._gf2 = kwargs.get("gf2", False)
self._allow_zero_inversions = bool(kwargs.get("allow_zero_inversions", False))
self._reverse_variables = bool(kwargs.get("reverse_variables", True))
with AllowZeroInversionsContext(self):
sub_byte_lookup = dict([(e, self.sub_byte(e)) for e in self._base])
self._sub_byte_lookup = sub_byte_lookup
if self._gf2:
self._polybori = kwargs.get("polybori", True)
def new_generator(self, **kwds):
r"""
Return a new ``SR`` instance equal to this instance
except for the parameters passed explicitly to this function.
INPUT:
- ``**kwds`` - see the ``SR`` constructor for accepted
parameters
EXAMPLE::
sage: sr = mq.SR(2,1,1,4); sr
SR(2,1,1,4)
sage: sr.ring().base_ring()
Finite Field in a of size 2^4
::
sage: sr2 = sr.new_generator(gf2=True); sr2
SR(2,1,1,4)
sage: sr2.ring().base_ring()
Finite Field of size 2
sage: sr3 = sr2.new_generator(correct_only=True)
sage: len(sr2.inversion_polynomials_single_sbox())
20
sage: len(sr3.inversion_polynomials_single_sbox())
19
"""
kwds.setdefault("n", self._n)
kwds.setdefault("r", self._r)
kwds.setdefault("c", self._c)
kwds.setdefault("e", self._e)
kwds.setdefault("star", self._star)
kwds.setdefault("postfix", self._postfix)
kwds.setdefault("order", self._order)
kwds.setdefault("allow_zero_inversions", self._allow_zero_inversions)
kwds.setdefault("aes_mode", self._aes_mode)
kwds.setdefault("gf2", self._gf2)
kwds.setdefault("reverse_variables", self._reverse_variables)
try:
polybori = self._polybori
except AttributeError:
polybori = False
kwds.setdefault("polybori", polybori)
try:
correct_only = self._correct_only
except AttributeError:
correct_only = False
kwds.setdefault("correct_only", correct_only)
try:
biaffine_only = self._biaffine_only
except AttributeError:
biaffine_only = False
kwds.setdefault("biaffine_only", biaffine_only)
if self._gf2 == kwds.get('gf2'):
return self.__class__(**kwds)
else:
return SR(**kwds)
def __getattr__(self, attr):
"""
EXAMPLE::
sage: sr = mq.SR(1, 2, 1, 4, gf2=True)
sage: sr.Mstar
[1 0 1 1 0 0 0 0]
[1 1 0 1 0 0 0 0]
[1 1 1 0 0 0 0 0]
[0 1 1 1 0 0 0 0]
[0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 0 1]
[0 0 0 0 1 1 1 0]
[0 0 0 0 0 1 1 1]
"""
if attr == "e":
return self._e
elif attr == "c":
return self._c
elif attr == "n":
return self._n
elif attr == "r":
return self._r
elif attr == "R":
self.R = self.ring()
return self.R
elif attr == "k":
self.k = self.base_ring()
return self.k
elif attr == "Lin":
self.Lin = self.lin_matrix()
return self.Lin
elif attr == "ShiftRows":
self.ShiftRows = self.shift_rows_matrix()
return self.ShiftRows
elif attr == "MixColumns":
self.MixColumns = self.mix_columns_matrix()
return self.MixColumns
elif attr == "M":
self.M = self.MixColumns * self.ShiftRows * self.Lin
return self.M
elif attr == "Mstar":
self.Mstar = self.ShiftRows * self.Lin
return self.Mstar
raise AttributeError, "%s has no attribute %s"%(type(self), attr)
def _repr_(self):
"""
EXAMPLES::
sage: sr = mq.SR(1, 2, 2, 4); sr #indirect doctest
SR(1,2,2,4)
sage: sr = mq.SR(1, 2, 2, 4, star=True); sr
SR*(1,2,2,4)
"""
if self._star:
return "SR*(%d,%d,%d,%d)"%(self._n, self._r, self._c, self._e)
else:
return "SR(%d,%d,%d,%d)"%(self._n, self._r, self._c, self._e)
def base_ring(self):
r"""
Return the base field of self as determined by
``self.e``.
EXAMPLE::
sage: sr = mq.SR(10, 2, 2, 4)
sage: sr.base_ring().polynomial()
a^4 + a + 1
The Rijndael polynomial::
sage: sr = mq.SR(10, 4, 4, 8)
sage: sr.base_ring().polynomial()
a^8 + a^4 + a^3 + a + 1
"""
try:
return self._base
except AttributeError:
if self._e == 4:
self._base = GF(2**4, 'a', modulus=(1, 1, 0, 0, 1))
elif self._e == 8:
self._base = GF(2**8, 'a', modulus=(1, 1, 0, 1, 1, 0, 0, 0, 1))
return self._base
def __cmp__(self, other):
"""
Two generators are considered equal if they agree on all parameters
passed to them during construction.
EXAMPLE::
sage: sr1 = mq.SR(2, 2, 2, 4)
sage: sr2 = mq.SR(2, 2, 2, 4)
sage: sr1 == sr2
True
::
sage: sr1 = mq.SR(2, 2, 2, 4)
sage: sr2 = mq.SR(2, 2, 2, 4, gf2=True)
sage: sr1 == sr2
False
"""
return cmp( (self.n, self.r, self.c, self.e, self._postfix, self._order, self._allow_zero_inversions, self._aes_mode, self._gf2, self._star ),
(other.n, other.r, other.c, other.e, other._postfix, other._order, other._allow_zero_inversions, other._aes_mode, other._gf2, other._star ) )
def sub_bytes(self, d):
r"""
Perform the non-linear transform on ``d``.
INPUT:
- ``d`` - state array or something coercible to a state array
EXAMPLE::
sage: sr = mq.SR(2, 1, 2, 8, gf2=True)
sage: k = sr.base_ring()
sage: A = Matrix(k, 1, 2 , [k(1), k.gen()])
sage: sr.sub_bytes(A)
[ a^6 + a^5 + a^4 + a^3 + a^2 a^6 + a^5 + a^4 + a^2 + a + 1]
"""
d = self.state_array(d)
return Matrix(self.base_ring(), d.nrows(), d.ncols(), [self.sub_byte(b) for b in d.list()])
def sub_byte(self, b):
r"""
Perform ``SubByte`` on a single byte/halfbyte ``b``.
A ``ZeroDivision`` exception is raised if an attempt is made
to perform an inversion on the zero element. This can be
disabled by passing ``allow_zero_inversion=True`` to the
constructor. A zero inversion can result in an inconsistent
equation system.
INPUT:
- ``b`` - an element in ``self.base_ring()``
EXAMPLE:
The S-Box table for `\GF{2^4}`::
sage: sr = mq.SR(1, 1, 1, 4, allow_zero_inversions=True)
sage: for e in sr.base_ring():
... print '% 20s % 20s'%(e, sr.sub_byte(e))
0 a^2 + a
a a^2 + 1
a^2 a
a^3 a^3 + 1
a + 1 a^2
a^2 + a a^2 + a + 1
a^3 + a^2 a + 1
a^3 + a + 1 a^3 + a^2
a^2 + 1 a^3 + a^2 + a
a^3 + a a^3 + a^2 + a + 1
a^2 + a + 1 a^3 + a
a^3 + a^2 + a 0
a^3 + a^2 + a + 1 a^3
a^3 + a^2 + 1 1
a^3 + 1 a^3 + a^2 + 1
1 a^3 + a + 1
"""
if not b:
if not self._allow_zero_inversions:
raise ZeroDivisionError, "A zero inversion occurred during an encryption or key schedule."
else:
return self.sbox_constant()
try:
return self._sub_byte_lookup[b]
except AttributeError:
e = self.e
k = self.k
b = b ** ( 2**e - 2 )
if e == 4:
if not hasattr(self, "_L"):
self._L = Matrix(GF(2), 4, 4, [[1, 1, 1, 0],
[0, 1, 1, 1],
[1, 0, 1, 1],
[1, 1, 0, 1]])
elif e==8:
if not hasattr(self, "_L"):
self._L = Matrix(GF(2), 8, 8, [[1, 0, 0, 0, 1, 1, 1, 1],
[1, 1, 0, 0, 0, 1, 1, 1],
[1, 1, 1, 0, 0, 0, 1, 1],
[1, 1, 1, 1, 0, 0, 0, 1],
[1, 1, 1, 1, 1, 0, 0, 0],
[0, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 0],
[0, 0, 0, 1, 1, 1, 1, 1]])
b = k(self._L * b._vector_())
if e == 4:
b = b + k.fetch_int(6)
elif e == 8:
b = b + k.fetch_int(99)
return b
def sbox_constant(self):
"""
Return the S-Box constant which is added after `L(x^{-1})` was
performed. That is ``0x63`` if ``e == 8`` or ``0x6`` if ``e ==
4``.
EXAMPLE::
sage: sr = mq.SR(10, 1, 1, 8)
sage: sr.sbox_constant()
a^6 + a^5 + a + 1
"""
k = self.k
if self.e == 4:
return k.fetch_int(6)
elif self.e == 8:
return k.fetch_int(99)
else:
raise TypeError, "sbox constant only defined for e in (4, 8)"
def sbox(self, inversion_only=False):
r"""
Return an S-Box object for this SR instance.
INPUT:
- ``inversion_only`` - do not include the `\GF{2}` affine map when
computing the S-Box (default: ``False``)
EXAMPLE::
sage: sr = mq.SR(1,2,2,4, allow_zero_inversions=True)
sage: S = sr.sbox(); S
(6, 11, 5, 4, 2, 14, 7, 10, 9, 13, 15, 12, 3, 1, 0, 8)
sage: sr.sub_byte(0)
a^2 + a
sage: sage_eval(str(sr.sub_byte(0)), {'a':2})
6
sage: S(0)
6
sage: sr.sub_byte(1)
a^3 + a + 1
sage: sage_eval(str(sr.sub_byte(1)), {'a':2})
11
sage: S(1)
11
sage: sr = mq.SR(1,2,2,8, allow_zero_inversions=True)
sage: S = sr.sbox(); S
(99, 124, 119, 123, 242, 107, 111, 197, 48, 1, 103, 43,
254, 215, 171, 118, 202, 130, 201, 125, 250, 89, 71, 240,
173, 212, 162, 175, 156, 164, 114, 192, 183, 253, 147, 38,
54, 63, 247, 204, 52, 165, 229, 241, 113, 216, 49, 21, 4,
199, 35, 195, 24, 150, 5, 154, 7, 18, 128, 226, 235, 39,
178, 117, 9, 131, 44, 26, 27, 110, 90, 160, 82, 59, 214,
179, 41, 227, 47, 132, 83, 209, 0, 237, 32, 252, 177, 91,
106, 203, 190, 57, 74, 76, 88, 207, 208, 239, 170, 251,
67, 77, 51, 133, 69, 249, 2, 127, 80, 60, 159, 168, 81,
163, 64, 143, 146, 157, 56, 245, 188, 182, 218, 33, 16,
255, 243, 210, 205, 12, 19, 236, 95, 151, 68, 23, 196,
167, 126, 61, 100, 93, 25, 115, 96, 129, 79, 220, 34, 42,
144, 136, 70, 238, 184, 20, 222, 94, 11, 219, 224, 50, 58,
10, 73, 6, 36, 92, 194, 211, 172, 98, 145, 149, 228, 121,
231, 200, 55, 109, 141, 213, 78, 169, 108, 86, 244, 234,
101, 122, 174, 8, 186, 120, 37, 46, 28, 166, 180, 198,
232, 221, 116, 31, 75, 189, 139, 138, 112, 62, 181, 102,
72, 3, 246, 14, 97, 53, 87, 185, 134, 193, 29, 158, 225,
248, 152, 17, 105, 217, 142, 148, 155, 30, 135, 233, 206,
85, 40, 223, 140, 161, 137, 13, 191, 230, 66, 104, 65,
153, 45, 15, 176, 84, 187, 22)
sage: sr.sub_byte(0)
a^6 + a^5 + a + 1
sage: sage_eval(str(sr.sub_byte(0)), {'a':2})
99
sage: S(0)
99
sage: sr.sub_byte(1)
a^6 + a^5 + a^4 + a^3 + a^2
sage: sage_eval(str(sr.sub_byte(1)), {'a':2})
124
sage: S(1)
124
sage: sr = mq.SR(1,2,2,4, allow_zero_inversions=True)
sage: S = sr.sbox(inversion_only=True); S
(0, 1, 9, 14, 13, 11, 7, 6, 15, 2, 12, 5, 10, 4, 3, 8)
sage: S(0)
0
sage: S(1)
1
sage: S(sr.k.gen())
a^3 + 1
"""
from sage.crypto.mq.sbox import SBox
k = self.base_ring()
if not inversion_only:
with AllowZeroInversionsContext(self):
S = [self.sub_byte(elem) for elem in sorted(k)]
return SBox(S)
else:
e = self.e
S = [elem ** (2**e - 2) for elem in sorted(k)]
return SBox(S)
def shift_rows(self, d):
r"""
Perform the ``ShiftRows`` operation on ``d``.
INPUT:
- ``d`` - state array or something coercible to a state array
EXAMPLES::
sage: sr = mq.SR(10, 4, 4, 4)
sage: E = sr.state_array() + 1; E
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
::
sage: sr.shift_rows(E)
[1 0 0 0]
[1 0 0 0]
[1 0 0 0]
[1 0 0 0]
"""
d = self.state_array(d)
ret = []
for i in range(d.nrows()):
ret += list(d.row(i)[i%d.ncols():]) + list(d.row(i)[:i%d.ncols()])
return Matrix(self.base_ring(), self._r, self._c, ret)
def mix_columns(self, d):
r"""
Perform the ``MixColumns`` operation on
``d``.
INPUT:
- ``d`` - state array or something coercible to a
state array
EXAMPLES::
sage: sr = mq.SR(10, 4, 4, 4)
sage: E = sr.state_array() + 1; E
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
::
sage: sr.mix_columns(E)
[ a a + 1 1 1]
[ 1 a a + 1 1]
[ 1 1 a a + 1]
[a + 1 1 1 a]
"""
d = self.state_array(d)
k = self.base_ring()
a = k.gen()
r = self._r
if r == 1:
M = Matrix(self.base_ring(), 1, 1, [[1]])
elif r == 2:
M = Matrix(self.base_ring(), 2, 2, [[a + 1, a],
[a, a + 1]])
elif r == 4:
M = Matrix(self.base_ring(), 4, 4, [[a, a+1, 1, 1],
[1, a, a+1, 1],
[1, 1, a, a+1],
[a+1, 1, 1, a]])
ret =[]
for column in d.columns():
ret.append(M * column)
return Matrix(k, d.ncols(), d.nrows(), ret).transpose()
def add_round_key(self, d, key):
r"""
Perform the ``AddRoundKey`` operation on
``d`` using ``key``.
INPUT:
- ``d`` - state array or something coercible to a
state array
- ``key`` - state array or something coercible to a
state array
EXAMPLE::
sage: sr = mq.SR(10, 4, 4, 4)
sage: D = sr.random_state_array()
sage: K = sr.random_state_array()
sage: sr.add_round_key(D, K) == K + D
True
"""
d = self.state_array(d)
key = self.state_array(key)
return d+key
def state_array(self, d=None):
"""
Convert the parameter to a state array.
INPUT:
- ``d`` - a matrix, a list, or a tuple (default: ``None``)
EXAMPLES::
sage: sr = mq.SR(2, 2, 2, 4)
sage: k = sr.base_ring()
sage: e1 = [k.fetch_int(e) for e in range(2*2)]; e1
[0, 1, a, a + 1]
sage: e2 = sr.phi( Matrix(k, 2*2, 1, e1) )
sage: sr.state_array(e1) # note the column major ordering
[ 0 a]
[ 1 a + 1]
sage: sr.state_array(e2)
[ 0 a]
[ 1 a + 1]
::
sage: sr.state_array()
[0 0]
[0 0]
"""
r = self.r
c = self.c
e = self.e
k = self.base_ring()
if d is None:
return Matrix(k, r, c)
if is_Matrix(d):
if d.nrows() == r*c*e:
return Matrix(k, c, r, self.antiphi(d).list()).transpose()
elif d.ncols() == c and d.nrows() == r and d.base_ring() == k:
return d
if isinstance(d, tuple([list, tuple])):
return Matrix(k, c, r, d).transpose()
def is_state_array(self, d):
"""
Return ``True`` if ``d`` is a state array, i.e. has the correct
dimensions and base field.
EXAMPLE::
sage: sr = mq.SR(2, 2, 4, 8)
sage: k = sr.base_ring()
sage: sr.is_state_array( matrix(k, 2, 4) )
True
::
sage: sr = mq.SR(2, 2, 4, 8)
sage: k = sr.base_ring()
sage: sr.is_state_array( matrix(k, 4, 4) )
False
"""
return is_Matrix(d) and \
d.nrows() == self.r and \
d.ncols() == self.c and \
d.base_ring() == self.base_ring()
def random_state_array(self, *args, **kwds):
r"""
Return a random element in ``MatrixSpace(self.base_ring(),
self.r, self.c)``.
EXAMPLE::
sage: sr = mq.SR(2, 2, 2, 4)
sage: sr.random_state_array()
[ a^2 a^3 + a + 1]
[a^3 + a^2 + a + 1 a + 1]
"""
return random_matrix(self.base_ring(), self._r, self._c, *args, **kwds)
def random_vector(self, *args, **kwds):
"""
Return a random vector as it might appear in the algebraic
expression of self.
EXAMPLE::
sage: sr = mq.SR(2, 2, 2, 4)
sage: sr.random_vector()
[ a^2]
[ a + 1]
[ a^2 + 1]
[ a]
[a^3 + a^2 + a + 1]
[ a^3 + a]
[ a^3]
[ a^3 + a^2]
[ a^3 + a + 1]
[ a^3 + 1]
[ a^3 + a^2 + 1]
[ a^3 + a^2 + a]
[ a + 1]
[ a^2 + 1]
[ a]
[ a^2]
.. note::
`\phi` was already applied to the result.
"""
return self.vector(self.random_state_array(*args, **kwds))
def random_element(self, elem_type = "vector", *args, **kwds):
"""
Return a random element for self. Other arguments and keywords are
passed to random_* methods.
INPUT:
- ``elem_type`` - either 'vector' or 'state array'
(default: ``'vector'``)
EXAMPLE::
sage: sr = mq.SR()
sage: sr.random_element()
[ a^2]
[ a + 1]
[a^2 + 1]
[ a]
sage: sr.random_element('state_array')
[a^3 + a + 1]
Passes extra positional or keyword arguments through::
sage: sr.random_element(density=0)
[0]
[0]
[0]
[0]
"""
if elem_type == "vector":
return self.random_vector(*args, **kwds)
elif elem_type == "state_array":
return self.random_state_array(*args, **kwds)
else:
raise TypeError, "parameter type not understood"
def key_schedule(self, kj, i):
"""
Return `k_i` for a given `i` and `k_j`
with `j = i-1`.
EXAMPLES::
sage: sr = mq.SR(10, 4, 4, 8, star=True, allow_zero_inversions=True)
sage: ki = sr.state_array()
sage: for i in range(10):
... ki = sr.key_schedule(ki, i+1)
sage: print sr.hex_str_matrix(ki)
B4 3E 23 6F
EF 92 E9 8F
5B E2 51 18
CB 11 CF 8E
"""
if i < 0:
raise TypeError, "i must be >= i"
if i == 0:
return kj
r = self.r
c = self.c
F = self.base_ring()
a = F.gen()
SubByte = self.sub_byte
rc = Matrix(F, r, c, ([a**(i-1)] * c) + [F(0)]*((r-1)*c) )
ki = Matrix(F, r, c)
if r == 1:
s0 = SubByte(kj[0, c-1])
if c > 1:
for q in range(c):
ki[0, q] = s0 + sum([kj[0, t] for t in range(q+1) ])
else:
ki[0, 0] = s0
elif r == 2:
s0 = SubByte(kj[1, c-1])
s1 = SubByte(kj[0, c-1])
if c > 1:
for q in range(c):
ki[0, q] = s0 + sum([ kj[0, t] for t in range(q+1) ])
ki[1, q] = s1 + sum([ kj[1, t] for t in range(q+1) ])
else:
ki[0, 0] = s0
ki[1, 0] = s1
elif r == 4:
if self._aes_mode:
s0 = SubByte(kj[1, c-1])
s1 = SubByte(kj[2, c-1])
s2 = SubByte(kj[3, c-1])
s3 = SubByte(kj[0, c-1])
else:
s0 = SubByte(kj[3, c-1])
s1 = SubByte(kj[2, c-1])
s2 = SubByte(kj[1, c-1])
s3 = SubByte(kj[0, c-1])
if c > 1:
for q in range(c):
ki[0, q] = s0 + sum([ kj[0, t] for t in range(q+1) ])
ki[1, q] = s1 + sum([ kj[1, t] for t in range(q+1) ])
ki[2, q] = s2 + sum([ kj[2, t] for t in range(q+1) ])
ki[3, q] = s3 + sum([ kj[3, t] for t in range(q+1) ])
else:
ki[0, 0] = s0
ki[1, 0] = s1
ki[2, 0] = s2
ki[3, 0] = s3
ki += rc
return ki
def __call__(self, P, K):
r"""
Encrypts the plaintext `P` using the key `K`.
Both must be given as state arrays or coercible to state arrays.
INPUTS:
- ``P`` - plaintext as state array or something coercible to a
qstate array
- ``K`` - key as state array or something coercible to a state
array
TESTS: The official AES test vectors::
sage: sr = mq.SR(10, 4, 4, 8, star=True, allow_zero_inversions=True)
sage: k = sr.base_ring()
sage: plaintext = sr.state_array([k.fetch_int(e) for e in range(16)])
sage: key = sr.state_array([k.fetch_int(e) for e in range(16)])
sage: print sr.hex_str_matrix( sr(plaintext, key) )
0A 41 F1 C6
94 6E C3 53
0B F0 94 EA
B5 45 58 5A
Brian Gladman's development vectors (dev_vec.txt)::
sage: sr = mq.SR(10, 4, 4, 8, star=True, allow_zero_inversions=True, aes_mode=True)
sage: k = sr.base_ring()
sage: plain = '3243f6a8885a308d313198a2e0370734'
sage: key = '2b7e151628aed2a6abf7158809cf4f3c'
sage: set_verbose(2)
sage: cipher = sr(plain, key)
R[01].start 193DE3BEA0F4E22B9AC68D2AE9F84808
R[01].s_box D42711AEE0BF98F1B8B45DE51E415230
R[01].s_row D4BF5D30E0B452AEB84111F11E2798E5
R[01].m_col 046681E5E0CB199A48F8D37A2806264C
R[01].k_sch A0FAFE1788542CB123A339392A6C7605
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R[02].s_box 49DED28945DB96F17F39871A7702533B
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R[02].m_col 584DCAF11B4B5AACDBE7CAA81B6BB0E5
R[02].k_sch F2C295F27A96B9435935807A7359F67F
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R[03].s_row ACC1D6B8EFB55A7B1323CFDF457311B5
R[03].m_col 75EC0993200B633353C0CF7CBB25D0DC
R[03].k_sch 3D80477D4716FE3E1E237E446D7A883B
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R[04].s_box 52502F2885A45ED7E311C807F6CF6A94
R[04].s_row 52A4C89485116A28E3CF2FD7F6505E07
R[04].m_col 0FD6DAA9603138BF6FC0106B5EB31301
R[04].k_sch EF44A541A8525B7FB671253BDB0BAD00
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R[05].s_box E14FD29BE8FBFBBA35C89653976CAE7C
R[05].s_row E1FB967CE8C8AE9B356CD2BA974FFB53
R[05].m_col 25D1A9ADBD11D168B63A338E4C4CC0B0
R[05].k_sch D4D1C6F87C839D87CAF2B8BC11F915BC
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R[06].s_row A14F3DFE78E803FC10D5A8DF4C632923
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R[07].s_box F7AB31F02783A9FF9B4340D354B53D3F
R[07].s_row F783403F27433DF09BB531FF54ABA9D3
R[07].m_col 1415B5BF461615EC274656D7342AD843
R[07].k_sch 4E54F70E5F5FC9F384A64FB24EA6DC4F
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R[08].s_box BE832CC8D43B86C00AE1D44DDA64F2FE
R[08].s_row BE3BD4FED4E1F2C80A642CC0DA83864D
R[08].m_col 00512FD1B1C889FF54766DCDFA1B99EA
R[08].k_sch EAD27321B58DBAD2312BF5607F8D292F
R[09].start EA835CF00445332D655D98AD8596B0C5
R[09].s_box 87EC4A8CF26EC3D84D4C46959790E7A6
R[09].s_row 876E46A6F24CE78C4D904AD897ECC395
R[09].m_col 473794ED40D4E4A5A3703AA64C9F42BC
R[09].k_sch AC7766F319FADC2128D12941575C006E
R[10].s_box E9098972CB31075F3D327D94AF2E2CB5
R[10].s_row E9317DB5CB322C723D2E895FAF090794
R[10].k_sch D014F9A8C9EE2589E13F0CC8B6630CA6
R[10].output 3925841D02DC09FBDC118597196A0B32
sage: set_verbose(0)
"""
r,c,e = self.r,self.c,self.e
F = self.base_ring()
_type = self.state_array
if isinstance(P, str):
P = self.state_array([F.fetch_int(ZZ(P[i:i+2], 16)) for i in range(0, len(P), 2)])
if isinstance(K, str):
K = self.state_array([F.fetch_int(ZZ(K[i:i+2], 16)) for i in range(0, len(K), 2)])
if self.is_state_array(P) and self.is_state_array(K):
_type = self.state_array
elif self.is_vector(P) and self.is_vector(K):
_type = self.vector
elif isinstance(P, (list,tuple)) and isinstance(K, (list,tuple)):
if len(P) == len(K) == r*c:
_type = self.state_array
elif len(P) == len(K) == r*c*e:
_type = self.vector
else:
raise TypeError, "length %d or %d doesn't match either %d or %d"%(len(P),len(K),r*c,r*c*e)
else:
raise TypeError, "plaintext or key parameter not understood"
P = self.state_array(P)
K = self.state_array(K)
AddRoundKey = self.add_round_key
SubBytes = self.sub_bytes
MixColumns = self.mix_columns
ShiftRows = self.shift_rows
KeyExpansion = self.key_schedule
P = AddRoundKey(P, K)
for r in range(self._n-1):
if get_verbose() >= 2:
print "R[%02d].start %s"%(r+1, self.hex_str_vector(P))
P = SubBytes(P)
if get_verbose() >= 2:
print "R[%02d].s_box %s"%(r+1, self.hex_str_vector(P))
P = ShiftRows(P)
if get_verbose() >= 2:
print "R[%02d].s_row %s"%(r+1, self.hex_str_vector(P))
P = MixColumns(P)
if get_verbose() >= 2:
print "R[%02d].m_col %s"%(r+1, self.hex_str_vector(P))
K = KeyExpansion(K, r+1)
if get_verbose() >= 2:
print "R[%02d].k_sch %s"%(r+1, self.hex_str_vector(K))
P = AddRoundKey(P, K)
P = SubBytes(P)
if get_verbose() >= 2:
print "R[%02d].s_box %s"%(self.n, self.hex_str_vector(P))
P = ShiftRows(P)
if get_verbose() >= 2:
print "R[%02d].s_row %s"%(self.n, self.hex_str_vector(P))
if not self._star:
P = MixColumns(P)
if get_verbose() >= 2:
print "R[%02d].m_col %s"%(self.n, self.hex_str_vector(P))
K = KeyExpansion(K, self._n)
if get_verbose() >= 2:
print "R[%02d].k_sch %s"%(self.n, self.hex_str_vector(K))
P = AddRoundKey(P, K)
if get_verbose() >= 2:
print "R[%02d].output %s"%(self.n, self.hex_str_vector(P))
return _type(P)
def hex_str(self, M, typ="matrix"):
r"""
Return a hex string for the provided AES state array/matrix.
INPUT:
- ``M`` - state array
- ``typ`` - controls what to return, either 'matrix'
or 'vector' (default: ``'matrix'``)
EXAMPLE::
sage: sr = mq.SR(2, 2, 2, 4)
sage: k = sr.base_ring()
sage: A = matrix(k, 2, 2, [1, k.gen(), 0, k.gen()^2])
sage: sr.hex_str(A)
' 1 2 \n 0 4 \n'
::
sage: sr.hex_str(A, typ='vector')
'1024'
"""
if typ == "matrix":
return self.hex_str_matrix(M)
elif typ == "vector":
return self.hex_str_vector(M)
else:
raise TypeError, "parameter type must either be 'matrix' or 'vector'"
def hex_str_matrix(self, M):
r"""
Return a two-dimensional AES-like representation of the matrix M.
That is, show the finite field elements as hex strings.
INPUT:
- ``M`` - an AES state array
EXAMPLE::
sage: sr = mq.SR(2, 2, 2, 4)
sage: k = sr.base_ring()
sage: A = matrix(k, 2, 2, [1, k.gen(), 0, k.gen()^2])
sage: sr.hex_str_matrix(A)
' 1 2 \n 0 4 \n'
"""
e = M.base_ring().degree()
st = [""]
for x in range(M.nrows()):
for y in range(M.ncols()):
if e == 8:
st.append("%02X"%(int(str(M[x, y].int_repr()))))
else:
st.append("%X"%(int(str(M[x, y].int_repr()))))
st.append("\n")
return " ".join(st)
def hex_str_vector(self, M):
"""
Return a one-dimensional AES-like representation of the matrix M.
That is, show the finite field elements as hex strings.
INPUT:
- ``M`` - an AES state array
EXAMPLE::
sage: sr = mq.SR(2, 2, 2, 4)
sage: k = sr.base_ring()
sage: A = matrix(k, 2, 2, [1, k.gen(), 0, k.gen()^2])
sage: sr.hex_str_vector(A)
'1024'
"""
e = M.base_ring().degree()
st = [""]
for y in range(M.ncols()):
for x in range(M.nrows()):
if e == 8:
st.append("%02X"%(int(str(M[x, y].int_repr()))))
else:
st.append("%X"%(int(str(M[x, y].int_repr()))))
return "".join(st)
def _insert_matrix_into_matrix(self, dst, src, row, col):
"""
Insert matrix src into matrix dst starting at row and col.
INPUT:
- ``dst`` - a matrix
- ``src`` - a matrix
- ``row`` - offset row
- ``col`` - offset columns
EXAMPLE::
sage: sr = mq.SR(10, 4, 4, 4)
sage: a = sr.k.gen()
sage: A = sr.state_array() + 1; A
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: B = Matrix(sr.base_ring(), 2, 2, [0, a, a+1, a^2]); B
[ 0 a]
[a + 1 a^2]
sage: sr._insert_matrix_into_matrix(A, B, 1, 1)
[ 1 0 0 0]
[ 0 0 a 0]
[ 0 a + 1 a^2 0]
[ 0 0 0 1]
"""
for i in range(src.nrows()):
for j in range(src.ncols()):
dst[row+i, col+j] = src[i, j]
return dst
def varformatstr(self, name, n=None, rc=None, e=None):
"""
Return a format string which is understood by print et al.
If a numerical value is omitted, the default value of ``self``
is used. The numerical values (``n``, ``rc``, ``e``) are used
to determine the width of the respective fields in the format
string.
INPUT:
- ``name`` - name of the variable
- ``n`` - number of rounds (default: ``None``)
- ``rc`` - number of rows \* number of cols (default: ``None``)
- ``e`` - exponent of base field (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(1, 2, 2, 4)
sage: sr.varformatstr('x')
'x%01d%01d%01d'
sage: sr.varformatstr('x', n=1000)
'x%03d%03d%03d'
"""
if n is None:
n = self.n
if rc is None:
rc = self.r * self.c
if e is None:
e = self.e
l = str(max([ len(str(rc-1)), len(str(n-1)), len(str(e-1)) ] ))
if name not in ("k", "s"):
pf = self._postfix
else:
pf = ""
format_string = name + pf + "%0" + l + "d" + "%0" + l + "d" + "%0" + l + "d"
return format_string
def varstr(self, name, nr, rc, e):
"""
Return a string representing a variable for the small scale
AES subject to the given constraints.
INPUT:
- ``name`` - variable name
- ``nr`` - number of round to create variable strings for
- ``rc`` - row*column index in state array
- ``e`` - exponent of base field
EXAMPLE::
sage: sr = mq.SR(10, 1, 2, 4)
sage: sr.varstr('x', 2, 1, 1)
'x211'
"""
format_string = self.varformatstr(name, self.n, self.r*self.c, self.e)
return format_string % (nr, rc, e)
def varstrs(self, name, nr, rc = None, e = None):
"""
Return a list of strings representing variables in ``self``.
INPUT:
- ``name`` - variable name
- ``nr`` - number of round to create variable strings for
- ``rc`` - number of rows * number of columns in the state array (default: ``None``)
- ``e`` - exponent of base field (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(10, 1, 2, 4)
sage: sr.varstrs('x', 2)
('x200', 'x201', 'x202', 'x203', 'x210', 'x211', 'x212', 'x213')
"""
if rc is None:
rc = self.r * self.c
if e is None:
e = self.e
n = self._n
format_string = self.varformatstr(name, n, rc, e)
return tuple([format_string % (nr, rci, ei) for rci in range(rc) for ei in range(e)])
def vars(self, name, nr, rc=None, e=None):
"""
Return a list of variables in ``self``.
INPUT:
- ``name`` - variable name
- ``nr`` - number of round to create variable strings for
- ``rc`` - number of rounds * number of columns in the state array (default: ``None``)
- ``e`` - exponent of base field (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(10, 1, 2, 4)
sage: sr.vars('x', 2)
(x200, x201, x202, x203, x210, x211, x212, x213)
"""
gd = self.variable_dict()
return tuple([gd[e] for e in self.varstrs(name, nr, rc, e)])
def variable_dict(self):
"""
Return a dictionary to access variables in ``self.R`` by their
names.
EXAMPLE::
sage: sr = mq.SR(1,1,1,4)
sage: sr.variable_dict()
{'x101': x101, 'x100': x100, 'x103': x103, 'x102': x102,
's002': s002, 'w100': w100, 'w101': w101, 'w102': w102,
'w103': w103, 'k100': k100, 'k101': k101, 'k102': k102,
'k103': k103, 's003': s003, 's001': s001, 'k002': k002,
'k001': k001, 'k000': k000, 'k003': k003, 's000': s000}
sage: sr = mq.SR(1,1,1,4,gf2=True)
sage: sr.variable_dict()
{'x101': x101, 'x100': x100, 'x103': x103, 'x102': x102,
's002': s002, 'w100': w100, 'w101': w101, 'w102': w102,
'w103': w103, 'k100': k100, 'k101': k101, 'k102': k102,
'k103': k103, 's003': s003, 's001': s001, 'k002': k002,
'k001': k001, 'k000': k000, 'k003': k003, 's000': s000}
"""
try:
R,gd = self._variable_dict
if R is self.R:
return gd
else:
pass
except AttributeError:
pass
gd = self.R.gens_dict()
self._variable_dict = self.R,gd
return gd
def block_order(self):
"""
Return a block order for self where each round is a block.
EXAMPLE::
sage: sr = mq.SR(2, 1, 1, 4)
sage: sr.block_order()
Block term order with blocks:
(Degree lexicographic term order of length 16,
Degree lexicographic term order of length 16,
Degree lexicographic term order of length 4)
::
sage: P = sr.ring(order='block')
sage: print P.repr_long()
Polynomial Ring
Base Ring : Finite Field in a of size 2^4
Size : 36 Variables
Block 0 : Ordering : deglex
Names : k200, k201, k202, k203, x200, x201, x202, x203, w200, w201, w202, w203, s100, s101, s102, s103
Block 1 : Ordering : deglex
Names : k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, w103, s000, s001, s002, s003
Block 2 : Ordering : deglex
Names : k000, k001, k002, k003
"""
r = self.r
c = self.c
e = self.e
n = self.n
T = None
for _n in range(n):
T = TermOrder('deglex', r*e + 3*r*c*e ) + T
T += TermOrder('deglex', r*c*e)
return T
def ring(self, order=None, reverse_variables=None):
r"""
Construct a ring as a base ring for the polynomial system.
By default, variables are ordered in the reverse of their natural
ordering, i.e. the reverse of as they appear.
INPUT:
- ``order`` - a monomial ordering (default: ``None``)
- ``reverse_variables`` - reverse rounds of variables (default: ``True``)
The variable assignment is as follows:
- `k_{i,j,l}` - subkey round `i` word `j` conjugate/bit `l`
- `s_{i,j,l}` - subkey inverse round `i` word `j` conjugate/bit `l`
- `w_{i,j,l}` - inversion input round `i` word `j` conjugate/bit `l`
- `x_{i,j,l}` - inversion output round `i` word `j` conjugate/bit `l`
Note that the variables are ordered in column major ordering
in the state array and that the bits are ordered in little
endian ordering.
For example, if `x_{0,1,0}` is a variable over `\GF{2}` for
`r=2` and `c=2` then refers to the *most* significant bit of
the entry in the position (1,0) in the state array matrix.
EXAMPLE::
sage: sr = mq.SR(2, 1, 1, 4)
sage: P = sr.ring(order='block')
sage: print P.repr_long()
Polynomial Ring
Base Ring : Finite Field in a of size 2^4
Size : 36 Variables
Block 0 : Ordering : deglex
Names : k200, k201, k202, k203, x200, x201, x202, x203, w200, w201, w202, w203, s100, s101, s102, s103
Block 1 : Ordering : deglex
Names : k100, k101, k102, k103, x100, x101, x102, x103, w100, w101, w102, w103, s000, s001, s002, s003
Block 2 : Ordering : deglex
Names : k000, k001, k002, k003
"""
r = self.r
c = self.c
e = self.e
n = self.n
if not self._gf2:
k = self.base_ring()
else:
k = GF(2)
if order is not None:
self._order = order
if self._order == 'block':
self._order = self.block_order()
if reverse_variables is None:
reverse_variables = self._reverse_variables
if reverse_variables:
process = lambda x: reversed(x)
else:
process = lambda x: x
if reverse_variables:
names = []
else:
names = self.varstrs("k", 0, r*c, e)
for _n in process(xrange(n)):
names += self.varstrs("k", _n+1, r*c, e)
names += self.varstrs("x", _n+1, r*c, e)
names += self.varstrs("w", _n+1, r*c, e)
names += self.varstrs("s", _n, r, e)
if reverse_variables:
names += self.varstrs("k", 0, r*c, e)
if self._gf2 and self._polybori:
return BooleanPolynomialRing(2*n*r*c*e + (n+1)*r*c*e + n*r*e, names, order=self._order)
else:
return PolynomialRing(k, 2*n*r*c*e + (n+1)*r*c*e + n*r*e, names, order=self._order)
def round_polynomials(self, i, plaintext=None, ciphertext=None):
r"""
Return list of polynomials for a given round `i`.
If ``i == 0`` a plaintext must be provided, if ``i == n`` a
ciphertext must be provided.
INPUT:
- ``i`` - round number
- ``plaintext`` - optional plaintext (mandatory in
first round)
- ``ciphertext`` - optional ciphertext (mandatory in
last round)
OUTPUT: tuple
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 4)
sage: k = sr.base_ring()
sage: p = [k.random_element() for _ in range(sr.r*sr.c)]
sage: sr.round_polynomials(0, plaintext=p)
(w100 + k000 + (a^2 + 1), w101 + k001 + (a), w102 + k002 + (a^2), w103 + k003 + (a + 1))
"""
r = self._r
c = self._c
e = self._e
n = self._n
R = self.R
M = self.M
_vars = self.vars
if i == 0:
w1 = Matrix(R, r*c*e, 1, _vars("w", 1, r*c, e))
k0 = Matrix(R, r*c*e, 1, _vars("k", 0, r*c, e))
if isinstance(plaintext, (tuple, list)) and len(plaintext) == r*c:
plaintext = Matrix(R, r*c*e, 1, self.phi(plaintext))
return tuple((w1 + k0 + plaintext).list())
elif i>0 and i<=n:
if self._star and i == n:
M = self.Mstar
xj = Matrix(R, r*c*e, 1, _vars("x", i, r*c, e))
ki = Matrix(R, r*c*e, 1, _vars("k", i, r*c, e))
rcon = Matrix(R, r*c*e, 1, self.phi([self.sbox_constant()]*r*c))
if i < n:
wj = Matrix(R, r*c*e, 1, _vars("w", i+1, r*c, e))
if i == n:
if isinstance(ciphertext, (tuple, list)) and len(ciphertext) == r*c:
ciphertext = Matrix(R, r*c*e, 1, self.phi(ciphertext))
wj = ciphertext
lin = (wj + ki + M * xj + rcon).list()
wi = Matrix(R, r*c*e, 1, _vars("w", i, r*c, e))
xi = Matrix(R, r*c*e, 1, _vars("x", i, r*c, e))
sbox = []
sbox += self.inversion_polynomials(xi, wi, r*c*e)
sbox += self.field_polynomials("x", i)
sbox += self.field_polynomials("w", i)
return tuple(lin + sbox)
def key_schedule_polynomials(self, i):
"""
Return polynomials for the `i`-th round of the key
schedule.
INPUT:
- ``i`` - round (`0 \leq i \leq n`)
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 4, gf2=True, polybori=False)
The 0-th subkey is the user provided key, so only conjugacy
relations or field polynomials are added.::
sage: sr.key_schedule_polynomials(0)
(k000^2 + k000, k001^2 + k001, k002^2 + k002, k003^2 + k003)
The 1-th subkey is derived from the user provided key according to
the key schedule which is non-linear.::
sage: sr.key_schedule_polynomials(1)
(k100 + s000 + s002 + s003,
k101 + s000 + s001 + s003 + 1,
k102 + s000 + s001 + s002 + 1,
k103 + s001 + s002 + s003 + 1,
k100^2 + k100, k101^2 + k101, k102^2 + k102, k103^2 + k103,
s000^2 + s000, s001^2 + s001, s002^2 + s002, s003^2 + s003,
s000*k000 + s000*k003 + s001*k002 + s002*k001 + s003*k000,
s000*k000 + s000*k001 + s001*k000 + s001*k003 + s002*k002 + s003*k001,
s000*k001 + s000*k002 + s001*k000 + s001*k001 + s002*k000 + s002*k003 + s003*k002,
s000*k000 + s000*k001 + s000*k003 + s001*k001 + s002*k000 + s002*k002 + s003*k000 + k000,
s000*k002 + s001*k000 + s001*k001 + s001*k003 + s002*k001 + s003*k000 + s003*k002 + k001,
s000*k000 + s000*k001 + s000*k002 + s001*k002 + s002*k000 + s002*k001 + s002*k003 + s003*k001 + k002,
s000*k001 + s001*k000 + s001*k002 + s002*k000 + s003*k001 + s003*k003 + k003,
s000*k000 + s000*k002 + s000*k003 + s001*k000 + s001*k001 + s002*k002 + s003*k000 + s000,
s000*k001 + s000*k003 + s001*k001 + s001*k002 + s002*k000 + s002*k003 + s003*k001 + s001,
s000*k000 + s000*k002 + s001*k000 + s001*k002 + s001*k003 + s002*k000 + s002*k001 + s003*k002 + s002,
s000*k001 + s000*k002 + s001*k000 + s001*k003 + s002*k001 + s003*k003 + s003,
s000*k002 + s001*k001 + s002*k000 + s003*k003 + 1)
"""
R = self.R
r = self.r
e = self.e
c = self.c
k = self.k
a = k.gen()
if i < 0:
raise TypeError, "i must by >= 0"
if i == 0:
return tuple(self.field_polynomials("k", i, r*c))
else:
L = self.lin_matrix(r)
ki = Matrix(R, r*c*e, 1, self.vars("k", i , r*c, e))
kj = Matrix(R, r*c*e, 1, self.vars("k", i-1, r*c, e))
si = Matrix(R, r*e, 1, self.vars("s", i-1, r, e))
rc = Matrix(R, r*e, 1, self.phi([a**(i-1)] + [k(0)]*(r-1)) )
d = Matrix(R, r*e, 1, self.phi([self.sbox_constant()]*r) )
sbox = []
sbox += self.field_polynomials("k", i)
sbox += self.field_polynomials("s", i-1, r)
if r == 1:
sbox += self.inversion_polynomials(kj[(c - 1)*e:(c - 1)*e + e], si[0:e], e)
if r == 2:
sbox += self.inversion_polynomials( kj[(2*c -1)*e : (2*c -1)*e + e] , si[0:1*e], e )
sbox += self.inversion_polynomials( kj[(2*c -2)*e : (2*c -2)*e + e] , si[e:2*e], e )
if r == 4:
if self._aes_mode:
sbox += self.inversion_polynomials( kj[(4*c-3)*e : (4*c-3)*e + e] , si[0*e : 1*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-2)*e : (4*c-2)*e + e] , si[1*e : 2*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-1)*e : (4*c-1)*e + e] , si[2*e : 3*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-4)*e : (4*c-4)*e + e] , si[3*e : 4*e] , e )
else:
sbox += self.inversion_polynomials( kj[(4*c-1)*e : (4*c-1)*e + e] , si[0*e : 1*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-2)*e : (4*c-2)*e + e] , si[1*e : 2*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-3)*e : (4*c-3)*e + e] , si[2*e : 3*e] , e )
sbox += self.inversion_polynomials( kj[(4*c-4)*e : (4*c-4)*e + e] , si[3*e : 4*e] , e )
si = L * si + d + rc
Sum = Matrix(R, r*e, 1)
lin = []
if c > 1:
for q in range(c):
t = range(r*e*(q) , r*e*(q+1) )
Sum += kj.matrix_from_rows(t)
lin += (ki.matrix_from_rows(t) + si + Sum).list()
else:
lin += (ki + si).list()
return tuple(lin + sbox)
def polynomial_system(self, P=None, K=None, C=None):
"""
Return a polynomial system for this small scale AES variant for a
given plaintext-key pair.
If neither ``P``, ``K`` nor ``C`` are provided, a random pair
(``P``, ``K``) will be generated. If ``P`` and ``C`` are
provided no ``K`` needs to be provided.
INPUT:
- ``P`` - vector, list, or tuple (default: ``None``)
- ``K`` - vector, list, or tuple (default: ``None``)
- ``C`` - vector, list, or tuple (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 4, gf2=True, polybori=True)
sage: P = sr.vector([0, 0, 1, 0])
sage: K = sr.vector([1, 0, 0, 1])
sage: F, s = sr.polynomial_system(P, K)
This returns a polynomial system::
sage: F
Polynomial Sequence with 36 Polynomials in 20 Variables
and a solution::
sage: s # random -- maybe we need a better doctest here?
{k000: 1, k001: 0, k003: 1, k002: 0}
This solution is not the only solution that we can learn from the
Groebner basis of the system.
::
sage: F.groebner_basis()[-3:]
[k000 + 1, k001, k003 + 1]
In particular we have two solutions::
sage: len(F.ideal().variety())
2
In the following example we provide ``C`` explicitly::
sage: C = sr(P,K)
sage: F,s = sr.polynomial_system(P=P, C=C)
sage: F
Polynomial Sequence with 36 Polynomials in 20 Variables
Alternatively, we can use symbols for the ``P`` and
``C``. First, we have to create a polynomial ring::
sage: sr = mq.SR(1, 1, 1, 4, gf2=True, polybori=True)
sage: R = sr.R
sage: vn = sr.varstrs("P",0,1,4) + R.variable_names() + sr.varstrs("C",0,1,4)
sage: R = BooleanPolynomialRing(len(vn),vn)
sage: sr.R = R
Now, we can construct the purely symbolic equation system::
sage: C = sr.vars("C",0); C
(C000, C001, C002, C003)
sage: P = sr.vars("P",0)
sage: F,s = sr.polynomial_system(P=P,C=C)
sage: [(k,v) for k,v in sorted(s.iteritems())] # this can be ignored
[(k003, 1), (k002, 1), (k001, 0), (k000, 1)]
sage: F
Polynomial Sequence with 36 Polynomials in 28 Variables
sage: F.part(0)
(P000 + w100 + k000, P001 + w101 + k001, P002 + w102 + k002, P003 + w103 + k003)
sage: F.part(-2)
(k100 + x100 + x102 + x103 + C000, k101 + x100 + x101 + x103 + C001 + 1, ...)
We show that the (returned) key is a solution to the returned system::
sage: sr = mq.SR(3,4,4,8, star=True, gf2=True, polybori=True)
sage: F,s = sr.polynomial_system()
sage: F.subs(s).groebner_basis() # long time
Polynomial Sequence with 1248 Polynomials in 1248 Variables
"""
plaintext = P
key = K
ciphertext = C
system = []
n = self._n
data = []
R = self.R
r,c,e = self.r,self.c,self.e
for d in (plaintext, key, ciphertext):
if d is None:
data.append( None )
elif isinstance(d, (tuple, list)):
if isinstance(d[0], (int,long)):
d = map(GF(2),d)
if len(d) == r*c*e and (d[0].parent() is R or d[0].parent() == R):
data.append( Matrix(R,r*c*e,1,d) )
continue
try:
data.append( self.phi(self.state_array(d)) )
except ValueError:
data.append( self.vector(d) )
elif self.is_state_array(d):
data.append( self.phi(d) )
elif self.is_vector(d):
data.append( d )
else:
data.append( False )
plaintext, key, ciphertext = data
if plaintext is False:
raise TypeError, "type %s of P not understood"%(type(plaintext))
elif plaintext is None:
plaintext = self.random_element("vector")
if key is None:
key = self.random_element("vector")
elif key is False and ciphertext is False:
raise TypeError, "type %s of K not understood"%(type(key))
if ciphertext is None:
ciphertext = self(plaintext, key)
elif ciphertext is False:
raise TypeError, "type %s of C not understood"%(type(ciphertext))
for i in range(n+1):
system.append( self.round_polynomials(i, plaintext, ciphertext) )
system.append( self.key_schedule_polynomials(i) )
if key is not None:
K = dict(zip(self.vars("k", 0), key.list()))
else:
K = None
return PolynomialSequence(self.R, system), K
class SR_gf2n(SR_generic):
r"""
Small Scale Variants of the AES polynomial system constructor over
`\GF{2^n}`.
"""
def vector(self, d=None):
"""
Constructs a vector suitable for the algebraic representation of
SR, i.e. BES.
INPUT:
- ``d`` - values for vector, must be understood by ``self.phi`` (default:``None``)
EXAMPLE::
sage: sr = mq.SR()
sage: sr
SR(1,1,1,4)
sage: k = sr.base_ring()
sage: A = Matrix(k, 1, 1, [k.gen()])
sage: sr.vector(A)
[ a]
[ a^2]
[ a + 1]
[a^2 + 1]
"""
r = self.r
c = self.c
e = self.e
k = self.base_ring()
if d is None:
return Matrix(k, r*c*e, 1)
elif d.ncols() == c and d.nrows() == r and d.base_ring() == k:
return Matrix(k, r*c*e, 1, self.phi(d).transpose().list())
def is_vector(self, d):
"""
Return ``True`` if ``d`` can be used as a vector for ``self``.
EXAMPLE::
sage: sr = mq.SR()
sage: sr
SR(1,1,1,4)
sage: k = sr.base_ring()
sage: A = Matrix(k, 1, 1, [k.gen()])
sage: B = sr.vector(A)
sage: sr.is_vector(A)
False
sage: sr.is_vector(B)
True
"""
return is_Matrix(d) and \
d.nrows() == self.r*self.c*self.e and \
d.ncols() == 1 and \
d.base_ring() == self.base_ring()
def phi(self, l):
r"""
The operation `\phi` from [MR02]_
Projects state arrays to their algebraic representation.
INPUT:
- ``l`` - element to perform `\phi` on.
EXAMPLE::
sage: sr = mq.SR(2, 1, 2, 4)
sage: k = sr.base_ring()
sage: A = matrix(k, 1, 2, [k.gen(), 0] )
sage: sr.phi(A)
[ a 0]
[ a^2 0]
[ a + 1 0]
[a^2 + 1 0]
"""
ret = []
if is_Matrix(l):
for e in l.transpose().list():
ret += [e**(2**i) for i in range(self.e)]
else:
for e in l:
ret += [e**(2**i) for i in range(self.e)]
if isinstance(l, list):
return ret
elif isinstance(l, tuple):
return tuple(ret)
elif is_Matrix(l):
return Matrix(l.base_ring(), l.ncols(), l.nrows()*self.e, ret).transpose()
else:
raise TypeError
def antiphi(self, l):
"""
The operation `\phi^{-1}` from [MR02]_ or the inverse of ``self.phi``.
INPUT:
- ``l`` - a vector in the sense of
``self.is_vector``
EXAMPLE::
sage: sr = mq.SR()
sage: A = sr.random_state_array()
sage: A
[a^2]
sage: sr.antiphi(sr.phi(A)) == A
True
"""
if is_Matrix(l):
ret = [e for e in l.transpose().list()[0:-1:self.e]]
else:
ret = [e for e in l[0:-1:self.e]]
if isinstance(l, list):
return ret
elif isinstance(l, tuple):
return tuple(ret)
elif is_Matrix(l):
return Matrix(self.base_ring(), l.ncols(), l.nrows()/self.e, ret).transpose()
else:
raise TypeError
def shift_rows_matrix(self):
"""
Return the ``ShiftRows`` matrix.
EXAMPLE::
sage: sr = mq.SR(1, 2, 2, 4)
sage: s = sr.random_state_array()
sage: r1 = sr.shift_rows(s)
sage: r2 = sr.state_array( sr.shift_rows_matrix() * sr.vector(s) )
sage: r1 == r2
True
"""
e = self.e
r = self.r
c = self.c
k = self.base_ring()
bs = r*c*e
shift_rows = Matrix(k, bs, bs)
I = MatrixSpace(k, e, e)(1)
for x in range(0, c):
for y in range(0, r):
_r = ((x*r)+y) * e
_c = (((x*r)+((r+1)*y)) * e) % bs
self._insert_matrix_into_matrix(shift_rows, I, _r, _c)
return shift_rows
def lin_matrix(self, length = None):
"""
Return the ``Lin`` matrix.
If no ``length`` is provided, the standard state space size is
used. The key schedule calls this method with an explicit
length argument because only ``self.r`` S-Box applications are
performed in the key schedule.
INPUT:
- ``length`` - length of state space (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 4)
sage: sr.lin_matrix()
[ a^2 + 1 1 a^3 + a^2 a^2 + 1]
[ a a 1 a^3 + a^2 + a + 1]
[ a^3 + a a^2 a^2 1]
[ 1 a^3 a + 1 a + 1]
"""
r = self.r
c = self.c
e = self.e
k = self.k
if length is None:
length = r*c
lin = Matrix(self.base_ring(), length*e, length*e)
if e == 4:
l = [ k.fetch_int(x) for x in (5, 1, 12, 5) ]
for k in range( 0, length ):
for i in range(0, 4):
for j in range(0, 4):
lin[k*4+j, k*4+i] = l[(i-j)%4] ** (2**j)
elif e == 8:
l = [ k.fetch_int(x) for x in (5, 9, 249, 37, 244, 1, 181, 143) ]
for k in range( 0, length ):
for i in range(0, 8):
for j in range(0, 8):
lin[k*8+j, k*8+i] = l[(i-j)%8] ** (2**j)
return lin
def mix_columns_matrix(self):
"""
Return the ``MixColumns`` matrix.
EXAMPLE::
sage: sr = mq.SR(1, 2, 2, 4)
sage: s = sr.random_state_array()
sage: r1 = sr.mix_columns(s)
sage: r2 = sr.state_array(sr.mix_columns_matrix() * sr.vector(s))
sage: r1 == r2
True
"""
def D(b):
"""
Return the `e x e` matrix `D` with `b^i` along the
diagonal.
EXAMPLE::
sage: sr = mq.SR(1, 2, 1, 4)
sage: sr.mix_columns_matrix() # indirect doctest
[ a + 1 0 0 0 a 0 0 0]
[ 0 a^2 + 1 0 0 0 a^2 0 0]
[ 0 0 a 0 0 0 a + 1 0]
[ 0 0 0 a^2 0 0 0 a^2 + 1]
[ a 0 0 0 a + 1 0 0 0]
[ 0 a^2 0 0 0 a^2 + 1 0 0]
[ 0 0 a + 1 0 0 0 a 0]
[ 0 0 0 a^2 + 1 0 0 0 a^2]
"""
D = Matrix(self.base_ring(), self._e, self._e)
for i in range(self._e):
D[i, i] = b**(2**i)
return D
r = self.r
c = self.c
e = self.e
k = self.k
a = k.gen()
M = Matrix(k, r*e, r*e)
if r == 1:
self._insert_matrix_into_matrix(M, D(1), 0, 0)
elif r == 2:
self._insert_matrix_into_matrix(M, D(a+1), 0, 0)
self._insert_matrix_into_matrix(M, D(a+1), e, e)
self._insert_matrix_into_matrix(M, D(a), e, 0)
self._insert_matrix_into_matrix(M, D(a), 0, e)
elif r == 4:
self._insert_matrix_into_matrix(M, D(a), 0, 0)
self._insert_matrix_into_matrix(M, D(a), e, e)
self._insert_matrix_into_matrix(M, D(a), 2*e, 2*e)
self._insert_matrix_into_matrix(M, D(a), 3*e, 3*e)
self._insert_matrix_into_matrix(M, D(a+1), 0, e)
self._insert_matrix_into_matrix(M, D(a+1), e, 2*e)
self._insert_matrix_into_matrix(M, D(a+1), 2*e, 3*e)
self._insert_matrix_into_matrix(M, D(a+1), 3*e, 0)
self._insert_matrix_into_matrix(M, D(1), 0, 2*e)
self._insert_matrix_into_matrix(M, D(1), e, 3*e)
self._insert_matrix_into_matrix(M, D(1), 2*e, 0)
self._insert_matrix_into_matrix(M, D(1), 3*e, 1*e)
self._insert_matrix_into_matrix(M, D(1), 0, 3*e)
self._insert_matrix_into_matrix(M, D(1), e, 0)
self._insert_matrix_into_matrix(M, D(1), 2*e, 1*e)
self._insert_matrix_into_matrix(M, D(1), 3*e, 2*e)
mix_columns = Matrix(k, r*c*e, r*c*e)
for i in range(c):
self._insert_matrix_into_matrix(mix_columns, M, r*e*i, r*e*i)
return mix_columns
def inversion_polynomials(self, xi, wi, length):
"""
Return polynomials to represent the inversion in the AES S-Box.
INPUT:
- ``xi`` - output variables
- ``wi`` - input variables
- ``length`` - length of both lists
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 8)
sage: R = sr.ring()
sage: xi = Matrix(R, 8, 1, sr.vars('x', 1))
sage: wi = Matrix(R, 8, 1, sr.vars('w', 1))
sage: sr.inversion_polynomials(xi, wi, 8)
[x100*w100 + 1,
x101*w101 + 1,
x102*w102 + 1,
x103*w103 + 1,
x104*w104 + 1,
x105*w105 + 1,
x106*w106 + 1,
x107*w107 + 1]
"""
return [xi[j, 0]*wi[j, 0] + 1 for j in range(length)]
def field_polynomials(self, name, i, l=None):
"""
Return list of conjugacy polynomials for a given round ``i``
and name ``name``.
INPUT:
- ``name`` - variable name
- ``i`` - round number
- ``l`` - r\*c (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(3, 1, 1, 8)
sage: sr.field_polynomials('x', 2)
[x200^2 + x201,
x201^2 + x202,
x202^2 + x203,
x203^2 + x204,
x204^2 + x205,
x205^2 + x206,
x206^2 + x207,
x207^2 + x200]
"""
r = self._r
c = self._c
e = self._e
n = self._n
if l is None:
l = r*c
_vars = self.vars(name, i, l, e)
return [_vars[e*j+i]**2 - _vars[e*j+(i+1)%e] for j in range(l) for i in range(e)]
class SR_gf2(SR_generic):
def __init__(self, n=1, r=1, c=1, e=4, star=False, **kwargs):
r"""
Small Scale Variants of the AES polynomial system constructor over
`\GF{2}`. See help for SR.
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: sr
SR(1,1,1,4)
"""
SR_generic.__init__(self, n, r, c, e, star, **kwargs)
self._correct_only = kwargs.get("correct_only", False)
self._biaffine_only = kwargs.get("biaffine_only", True)
def vector(self, d=None):
"""
Constructs a vector suitable for the algebraic representation of
SR.
INPUT:
- ``d`` - values for vector (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: sr
SR(1,1,1,4)
sage: k = sr.base_ring()
sage: A = Matrix(k, 1, 1, [k.gen()])
sage: sr.vector(A)
[0]
[0]
[1]
[0]
"""
r = self.r
c = self.c
e = self.e
k = GF(2)
if d is None:
return Matrix(k, r*c*e, 1)
elif is_Matrix(d) and d.ncols() == c and d.nrows() == r and d.base_ring() == self.k:
l = flatten([self.phi(x) for x in d.transpose().list()], (Vector_modn_dense,list,tuple))
return Matrix(k, r*c*e, 1, l)
elif isinstance(d, (list, tuple)):
if len(d) == self.r*self.c:
l = flatten([self.phi(x) for x in d], (Vector_modn_dense,list,tuple))
return Matrix(k, r*c*e, 1, l)
elif len(d) == self.r*self.c*self.e:
return Matrix(k, r*c*e, 1, d)
else:
raise TypeError
else:
raise TypeError
def is_vector(self, d):
"""
Return ``True`` if the given matrix satisfies the conditions
for a vector as it appears in the algebraic expression of
``self``.
INPUT:
- ``d`` - matrix
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: sr
SR(1,1,1,4)
sage: k = sr.base_ring()
sage: A = Matrix(k, 1, 1, [k.gen()])
sage: B = sr.vector(A)
sage: sr.is_vector(A)
False
sage: sr.is_vector(B)
True
"""
return is_Matrix(d) and \
d.nrows() == self.r*self.c*self.e and \
d.ncols() == 1 and \
d.base_ring() == GF(2)
def phi(self, l, diffusion_matrix=False):
r"""
The operation `\phi` from [MR02]_
Given a list/matrix of elements in `\GF{2^e}`, return a
matching list/matrix of elements in `\GF{2}`.
INPUT:
- ``l`` - element to perform `\phi` on.
- ``diffusion_matrix`` - if ``True``, the given matrix ``l`` is
transformed to a matrix which performs the same operation
over `\GF{2}` as ``l`` over `\GF{2^n}` (default: ``False``).
EXAMPLE::
sage: sr = mq.SR(2, 1, 2, 4, gf2=True)
sage: k = sr.base_ring()
sage: A = matrix(k, 1, 2, [k.gen(), 0] )
sage: sr.phi(A)
[0 0]
[0 0]
[1 0]
[0 0]
"""
ret = []
r, c, e = self.r, self.c, self.e
if is_Matrix(l) and diffusion_matrix and \
l.nrows() == r*c and l.ncols() == r*c and \
l.base_ring() == self.k:
B = Matrix(GF(2), r*c*e, r*c*e)
for x in range(r*c):
for y in range(r*c):
T = self._mul_matrix(l[x, y])
self._insert_matrix_into_matrix(B, T, x*e, y*e)
return B
if l in self.k:
return list(reversed(l._vector_()))
if is_Matrix(l):
for x in l.transpose().list():
ret += list(reversed(x._vector_()))
else:
for x in l:
ret += list(reversed(x._vector_()))
if isinstance(l, list):
return ret
elif isinstance(l, tuple):
return tuple(ret)
elif is_Matrix(l):
return Matrix(GF(2), l.ncols(), l.nrows()*self.e, ret).transpose()
else: raise TypeError
def antiphi(self, l):
"""
The operation `\phi^{-1}` from [MR02]_ or the inverse of ``self.phi``.
INPUT:
- ``l`` - a vector in the sense of ``self.is_vector``
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: A = sr.random_state_array()
sage: A
[a^2]
sage: sr.antiphi(sr.phi(A)) == A
True
"""
e = self.e
V = self.k.vector_space()
if is_Matrix(l):
l2 = l.transpose().list()
else:
l2 = l
ret = []
for i in range(0, len(l2), e):
ret.append( self.k(V(list(reversed(l2[i:i+e])))) )
if isinstance(l, list):
return ret
elif isinstance(l, tuple):
return tuple(ret)
elif is_Matrix(l):
return Matrix(self.base_ring(), self.r *self.c, 1, ret)
else:
raise TypeError
def shift_rows_matrix(self):
"""
Return the ``ShiftRows`` matrix.
EXAMPLE::
sage: sr = mq.SR(1, 2, 2, 4, gf2=True)
sage: s = sr.random_state_array()
sage: r1 = sr.shift_rows(s)
sage: r2 = sr.state_array( sr.shift_rows_matrix() * sr.vector(s) )
sage: r1 == r2
True
"""
r = self.r
c = self.c
k = self.k
bs = r*c
shift_rows = Matrix(k, r*c, r*c)
for x in range(0, c):
for y in range(0, r):
_r = ((x*r)+y)
_c = ((x*r)+((r+1)*y)) % bs
shift_rows[_r, _c] = 1
return self.phi(shift_rows, diffusion_matrix=True)
def mix_columns_matrix(self):
"""
Return the ``MixColumns`` matrix.
EXAMPLE::
sage: sr = mq.SR(1, 2, 2, 4, gf2=True)
sage: s = sr.random_state_array()
sage: r1 = sr.mix_columns(s)
sage: r2 = sr.state_array(sr.mix_columns_matrix() * sr.vector(s))
sage: r1 == r2
True
"""
r = self.r
c = self.c
k = self.k
a = k.gen()
if r == 1:
M = Matrix(k, r, r, 1)
elif r == 2:
M = Matrix(k, r, r, [a+1, a, a, a+1])
elif r == 4:
M = Matrix(k, r, [a, a+1, 1, 1, \
1, a, a+1, 1, \
1, 1, a, a+1, \
a+1, 1, 1, a])
mix_columns = Matrix(k, r*c, r*c)
for i in range(c):
self._insert_matrix_into_matrix(mix_columns, M, r*i, r*i)
return self.phi(mix_columns, diffusion_matrix=True)
def lin_matrix(self, length=None):
"""
Return the ``Lin`` matrix.
If no ``length`` is provided, the standard state space size is
used. The key schedule calls this method with an explicit
length argument because only ``self.r`` S-Box applications are
performed in the key schedule.
INPUT:
- ``length`` - length of state space (default: ``None``)
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 4, gf2=True)
sage: sr.lin_matrix()
[1 0 1 1]
[1 1 0 1]
[1 1 1 0]
[0 1 1 1]
"""
r, c, e = self.r, self.c, self.e
if length is None:
length = r*c
if e == 8:
Z = Matrix(GF(2), 8, 8, [1, 0, 0, 0, 1, 1, 1, 1, \
1, 1, 0, 0, 0, 1, 1, 1, \
1, 1, 1, 0, 0, 0, 1, 1, \
1, 1, 1, 1, 0, 0, 0, 1, \
1, 1, 1, 1, 1, 0, 0, 0, \
0, 1, 1, 1, 1, 1, 0, 0, \
0, 0, 1, 1, 1, 1, 1, 0, \
0, 0, 0, 1, 1, 1, 1, 1])
else:
Z = Matrix(GF(2), 4, 4, [1, 1, 1, 0, \
0, 1, 1, 1, \
1, 0, 1, 1, \
1, 1, 0, 1])
Z = Z.transpose()
lin = Matrix(GF(2), length*e, length*e)
for i in range(length):
self._insert_matrix_into_matrix(lin, Z, i*e, i*e)
return lin
def _mul_matrix(self, x):
r"""
Given an element `x` in self.base_ring(), return a matrix
which performs the same operation on a when interpreted over
`\GF{2^e}` as `x` over `\GF{2^e}`.
INPUT:
- ``x`` - an element in self.base_ring()
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: a = sr.k.gen()
sage: A = sr._mul_matrix(a^2+1)
sage: sr.antiphi( A * sr.vector([a+1]) )
[a^3 + a^2 + a + 1]
::
sage: (a^2 + 1)*(a+1)
a^3 + a^2 + a + 1
"""
a = self.k.gen()
k = self.k
e = self.e
a = k.gen()
columns = []
for i in reversed(range(e)):
columns.append( list(reversed((x * a**i)._vector_())) )
return Matrix(GF(2), e, e, columns).transpose()
def _square_matrix(self):
"""
Return a matrix of dimension self.e x self.e which performs the
squaring operation over `GF(2^n)` on vectors of length e.
EXAMPLE::
sage: sr = mq.SR(gf2=True)
sage: a = sr.k.gen()
sage: S = sr._square_matrix()
sage: sr.antiphi( S * sr.vector([a^3+1]) )
[a^3 + a^2 + 1]
::
sage: (a^3 + 1)^2
a^3 + a^2 + 1
"""
a = self.k.gen()
e = self.e
columns = []
for i in reversed(range(e)):
columns.append( list(reversed(((a**i)**2)._vector_())) )
return Matrix(GF(2), e , e, columns).transpose()
def inversion_polynomials_single_sbox(self, x=None, w=None, biaffine_only=None, correct_only=None):
"""
Return inversion polynomials of a single S-Box.
INPUT:
- ``xi`` - output variables
- ``wi`` - input variables
- ``length`` - length of both lists
EXAMPLES::
sage: sr = mq.SR(1, 1, 1, 8, gf2=True)
sage: len(sr.inversion_polynomials_single_sbox())
24
sage: len(sr.inversion_polynomials_single_sbox(correct_only=True))
23
sage: len(sr.inversion_polynomials_single_sbox(biaffine_only=False))
40
sage: len(sr.inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True))
39
sage: sr = mq.SR(1, 1, 1, 8, gf2=True)
sage: l0 = sr.inversion_polynomials_single_sbox(); len(l0)
24
sage: l1 = sr.inversion_polynomials_single_sbox(correct_only=True); len(l1)
23
sage: l2 = sr.inversion_polynomials_single_sbox(biaffine_only=False); len(l2)
40
sage: l3 = sr.inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True); len(l3)
39
sage: set(l0) == set(sr._inversion_polynomials_single_sbox())
True
sage: set(l1) == set(sr._inversion_polynomials_single_sbox(correct_only=True))
True
sage: set(l2) == set(sr._inversion_polynomials_single_sbox(biaffine_only=False))
True
sage: set(l3) == set(sr._inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True))
True
sage: sr = mq.SR(1, 1, 1, 4, gf2=True)
sage: l0 = sr.inversion_polynomials_single_sbox(); len(l0)
12
sage: l1 = sr.inversion_polynomials_single_sbox(correct_only=True); len(l1)
11
sage: l2 = sr.inversion_polynomials_single_sbox(biaffine_only=False); len(l2)
20
sage: l3 = sr.inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True); len(l3)
19
sage: set(l0) == set(sr._inversion_polynomials_single_sbox())
True
sage: set(l1) == set(sr._inversion_polynomials_single_sbox(correct_only=True))
True
sage: set(l2) == set(sr._inversion_polynomials_single_sbox(biaffine_only=False))
True
sage: set(l3) == set(sr._inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True))
True
"""
e = self.e
if biaffine_only is None:
biaffine_only = self._biaffine_only
if correct_only is None:
correct_only = self._correct_only
if x is None and w is None:
names = ["w%d" % i for i in reversed(range(e))] + ["x%d"%i for i in reversed(range(e))]
P = PolynomialRing(GF(2), e*2, names, order='lex')
x = P.gens()[e:]
w = P.gens()[:e]
else:
if isinstance(x, (tuple, list)):
P = x[0].parent()
elif is_Matrix(x):
P = x.base_ring()
else:
raise TypeError, "x not understood"
if is_Matrix(x):
x = x.column(0).list()
if is_Matrix(w):
w = w.column(0).list()
if e == 4:
w3,w2,w1,w0 = w
x3,x2,x1,x0 = x
l = [w3*x3 + w3*x0 + w2*x1 + w1*x2 + w0*x3,
w3*x3 + w3*x2 + w2*x3 + w2*x0 + w1*x1 + w0*x2,
w3*x2 + w3*x1 + w2*x3 + w2*x2 + w1*x3 + w1*x0 + w0*x1,
w3*x3 + w3*x2 + w3*x0 + w2*x2 + w1*x3 + w1*x1 + w0*x3 + x3,
w3*x1 + w2*x3 + w2*x2 + w2*x0 + w1*x2 + w0*x3 + w0*x1 + x2,
w3*x3 + w3*x2 + w3*x1 + w2*x1 + w1*x3 + w1*x2 + w1*x0 + w0*x2 + x1,
w3*x2 + w2*x3 + w2*x1 + w1*x3 + w0*x2 + w0*x0 + x0,
w3*x3 + w3*x1 + w3*x0 + w3 + w2*x3 + w2*x2 + w1*x1 + w0*x3,
w3*x2 + w3*x0 + w2*x2 + w2*x1 + w2 + w1*x3 + w1*x0 + w0*x2,
w3*x3 + w3*x1 + w2*x3 + w2*x1 + w2*x0 + w1*x3 + w1*x2 + w1 + w0*x1,
w3*x2 + w3*x1 + w2*x3 + w2*x0 + w1*x2 + w0*x0 + w0]
if not correct_only:
l.append(w3*x1 + w2*x2 + w1*x3 + w0*x0 + 1)
if not biaffine_only:
l.extend([w3*x2 + w3*x1 + w3*x0 + w2*x3 + w2*x1 + w1*x3 + w1*x2 + w0*x3 + x3**2 + x3*x2 + x3*x1 + x2**2 + x1**2,
w3*x2 + w2*x2 + w2*x1 + w2*x0 + w1*x3 + w1*x1 + w0*x3 + w0*x2 + x3*x2 + x3*x1 + x3*x0 + x2**2 + x2*x1 + x2*x0 + x1*x0,
w3*x2 + w3*x1 + w2*x2 + w1*x2 + w1*x1 + w1*x0 + w0*x3 + w0*x1 + x3**2 + x3*x2 + x2*x0 + x1*x0,
w3*x3 + w3*x1 + w2*x3 + w2*x2 + w1*x3 + w0*x3 + w0*x2 + w0*x1 + w0*x0 + x3*x1 + x2*x1 + x2*x0 + x0**2,
w3**2 + w3*w2 + w3*w1 + w3*x2 + w3*x1 + w3*x0 + w2**2 + w2*x3 + w2*x1 + w1**2 + w1*x3 + w1*x2 + w0*x3,
w3*w2 + w3*w1 + w3*w0 + w3*x1 + w3*x0 + w2**2 + w2*w1 + w2*w0 + w2*x3 + w2*x2 + w2*x0 + w1*w0 + w1*x2 + w1*x1 + w0*x2,
w3**2 + w3*w2 + w3*x0 + w2*w0 + w2*x3 + w2*x2 + w2*x1 + w1*w0 + w1*x3 + w1*x1 + w1*x0 + w0*x1,
w3*w1 + w3*x3 + w3*x2 + w3*x1 + w3*x0 + w2*w1 + w2*w0 + w2*x2 + w2*x0 + w1*x3 + w1*x0 + w0**2 + w0*x0])
return l
else:
w7,w6,w5,w4,w3,w2,w1,w0 = w
x7,x6,x5,x4,x3,x2,x1,x0 = x
l = [w7*x7 + w7*x5 + w7*x4 + w7*x0 + w6*x6 + w6*x5 + w6*x1 + w5*x7 + w5*x6 + w5*x2 + w4*x7 + w4*x3 + w3*x4 + w2*x5 + w1*x6 + w0*x7,
w7*x6 + w7*x4 + w7*x3 + w6*x7 + w6*x5 + w6*x4 + w6*x0 + w5*x6 + w5*x5 + w5*x1 + w4*x7 + w4*x6 + w4*x2 + w3*x7 + w3*x3 + w2*x4 + w1*x5 + w0*x6,
w7*x5 + w7*x3 + w7*x2 + w6*x6 + w6*x4 + w6*x3 + w5*x7 + w5*x5 + w5*x4 + w5*x0 + w4*x6 + w4*x5 + w4*x1 + w3*x7 + w3*x6 + w3*x2 + w2*x7 + w2*x3 + w1*x4 + w0*x5,
w7*x7 + w7*x4 + w7*x2 + w7*x1 + w6*x5 + w6*x3 + w6*x2 + w5*x6 + w5*x4 + w5*x3 + w4*x7 + w4*x5 + w4*x4 + w4*x0 + w3*x6 + w3*x5 + w3*x1 + w2*x7 + w2*x6 + w2*x2 + w1*x7 + w1*x3 + w0*x4,
w7*x7 + w7*x6 + w7*x5 + w7*x4 + w7*x3 + w7*x1 + w6*x7 + w6*x6 + w6*x5 + w6*x4 + w6*x2 + w5*x7 + w5*x6 + w5*x5 + w5*x3 + w4*x7 + w4*x6 + w4*x4 + w3*x7 + w3*x5 + w3*x0 + w2*x6 + w2*x1 \
+ w1*x7 + w1*x2 + w0*x3,
w7*x6 + w7*x3 + w7*x2 + w6*x7 + w6*x4 + w6*x3 + w5*x5 + w5*x4 + w4*x6 + w4*x5 + w3*x7 + w3*x6 + w2*x7 + w2*x0 + w1*x1 + w0*x2,
w7*x7 + w7*x5 + w7*x2 + w7*x1 + w6*x6 + w6*x3 + w6*x2 + w5*x7 + w5*x4 + w5*x3 + w4*x5 + w4*x4 + w3*x6 + w3*x5 + w2*x7 + w2*x6 + w1*x7 + w1*x0 + w0*x1,
w7*x6 + w7*x5 + w7*x2 + w7*x0 + w6*x7 + w6*x4 + w6*x3 + w5*x7 + w5*x6 + w5*x3 + w5*x1 + w4*x5 + w4*x4 + w3*x7 + w3*x4 + w3*x2 + w2*x6 + w2*x5 + w1*x5 + w1*x3 + w0*x7 + w0*x6 + x7,
w7*x6 + w7*x3 + w7*x2 + w6*x6 + w6*x5 + w6*x2 + w6*x0 + w5*x7 + w5*x4 + w5*x3 + w4*x7 + w4*x6 + w4*x3 + w4*x1 + w3*x5 + w3*x4 + w2*x7 + w2*x4 + w2*x2 + w1*x6 + w1*x5 + w0*x5 + w0*x3 \
+ x6,
w7*x7 + w7*x5 + w7*x4 + w7*x1 + w6*x6 + w6*x3 + w6*x2 + w5*x6 + w5*x5 + w5*x2 + w5*x0 + w4*x7 + w4*x4 + w4*x3 + w3*x7 + w3*x6 + w3*x3 + w3*x1 + w2*x5 + w2*x4 + w1*x7 + w1*x4 + w1*x2 \
+ w0*x6 + w0*x5 + x5,
w7*x7 + w7*x5 + w7*x2 + w7*x1 + w6*x7 + w6*x5 + w6*x4 + w6*x1 + w5*x6 + w5*x3 + w5*x2 + w4*x6 + w4*x5 + w4*x2 + w4*x0 + w3*x7 + w3*x4 + w3*x3 + w2*x7 + w2*x6 + w2*x3 + w2*x1 + w1*x5 \
+ w1*x4 + w0*x7 + w0*x4 + w0*x2 + x4,
w7*x5 + w7*x4 + w7*x3 + w7*x2 + w6*x5 + w6*x4 + w6*x3 + w6*x2 + w6*x1 + w5*x6 + w5*x5 + w5*x4 + w5*x3 + w4*x6 + w4*x5 + w4*x4 + w4*x3 + w4*x2 + w3*x7 + w3*x6 + w3*x5 + w3*x4 + w3*x0 \
+ w2*x7 + w2*x6 + w2*x5 + w2*x4 + w2*x3 + w1*x7 + w1*x6 + w1*x5 + w1*x1 + w0*x7 + w0*x6 + w0*x5 + w0*x4 + x3,
w7*x7 + w7*x6 + w7*x5 + w7*x4 + w7*x3 + w7*x1 + w6*x7 + w6*x5 + w6*x2 + w5*x7 + w5*x6 + w5*x5 + w5*x4 + w5*x2 + w4*x6 + w4*x3 + w3*x7 + w3*x6 + w3*x5 + w3*x3 + w2*x7 + w2*x4 + w2*x0 \
+ w1*x7 + w1*x6 + w1*x4 + w0*x5 + w0*x1 + x2,
w7*x6 + w7*x4 + w7*x1 + w6*x7 + w6*x6 + w6*x5 + w6*x4 + w6*x3 + w6*x1 + w5*x7 + w5*x5 + w5*x2 + w4*x7 + w4*x6 + w4*x5 + w4*x4 + w4*x2 + w3*x6 + w3*x3 + w2*x7 + w2*x6 + w2*x5 + w2*x3 \
+ w1*x7 + w1*x4 + w1*x0 + w0*x7 + w0*x6 + w0*x4 + x1,
w7*x7 + w7*x4 + w7*x3 + w6*x7 + w6*x6 + w6*x3 + w6*x1 + w5*x5 + w5*x4 + w4*x7 + w4*x4 + w4*x2 + w3*x6 + w3*x5 + w2*x5 + w2*x3 + w1*x7 + w1*x6 + w0*x6 + w0*x4 + w0*x0 + x0,
w7*x6 + w7*x5 + w7*x3 + w7*x0 + w7 + w6*x7 + w6*x5 + w6*x2 + w6*x0 + w5*x7 + w5*x4 + w5*x2 + w5*x1 + w4*x6 + w4*x4 + w4*x3 + w3*x6 + w3*x5 + w3*x1 + w2*x7 + w2*x3 + w1*x5 + w0*x7,
w7*x5 + w7*x4 + w7*x2 + w6*x7 + w6*x6 + w6*x4 + w6*x1 + w6 + w5*x6 + w5*x3 + w5*x1 + w5*x0 + w4*x5 + w4*x3 + w4*x2 + w3*x7 + w3*x5 + w3*x4 + w3*x0 + w2*x7 + w2*x6 + w2*x2 + w1*x4 \
+ w0*x6,
w7*x7 + w7*x4 + w7*x3 + w7*x1 + w6*x6 + w6*x5 + w6*x3 + w6*x0 + w5*x7 + w5*x5 + w5*x2 + w5*x0 + w5 + w4*x7 + w4*x4 + w4*x2 + w4*x1 + w3*x6 + w3*x4 + w3*x3 + w2*x6 + w2*x5 + w2*x1 \
+ w1*x7 + w1*x3 + w0*x5,
w7*x7 + w7*x6 + w7*x3 + w7*x2 + w7*x0 + w6*x5 + w6*x4 + w6*x2 + w5*x7 + w5*x6 + w5*x4 + w5*x1 + w4*x6 + w4*x3 + w4*x1 + w4*x0 + w4 + w3*x5 + w3*x3 + w3*x2 + w2*x7 + w2*x5 + w2*x4 \
+ w2*x0 + w1*x7 + w1*x6 + w1*x2 + w0*x4,
w7*x3 + w7*x2 + w7*x1 + w7*x0 + w6*x5 + w6*x4 + w6*x3 + w6*x2 + w6*x1 + w6*x0 + w5*x7 + w5*x6 + w5*x5 + w5*x4 + w5*x3 + w5*x2 + w5*x1 + w5*x0 + w4*x7 + w4*x6 + w4*x5 + w4*x4 \
+ w4*x3 + w4*x2 + w4*x0 + w3*x7 + w3*x6 + w3*x5 + w3*x4 + w3*x2 + w3 + w2*x7 + w2*x6 + w2*x4 + w1*x6 + w1*x1 + w0*x3,
w7*x7 + w7*x6 + w7*x5 + w7*x3 + w7*x2 + w7*x1 + w6*x7 + w6*x5 + w6*x4 + w6*x3 + w6*x1 + w5*x7 + w5*x6 + w5*x5 + w5*x3 + w5*x0 + w4*x7 + w4*x5 + w4*x2 + w4*x1 + w3*x7 + w3*x4 \
+ w3*x3 + w2*x6 + w2*x5 + w2 + w1*x7 + w1*x0 + w0*x2,
w7*x6 + w7*x5 + w7*x4 + w7*x2 + w7*x1 + w7*x0 + w6*x7 + w6*x6 + w6*x4 + w6*x3 + w6*x2 + w6*x0 + w5*x6 + w5*x5 + w5*x4 + w5*x2 + w4*x7 + w4*x6 + w4*x4 + w4*x1 + w4*x0 + w3*x6 \
+ w3*x3 + w3*x2 + w2*x5 + w2*x4 + w1*x7 + w1*x6 + w1 + w0*x1,
w7*x7 + w7*x6 + w7*x4 + w7*x1 + w6*x6 + w6*x3 + w6*x1 + w6*x0 + w5*x5 + w5*x3 + w5*x2 + w4*x7 + w4*x5 + w4*x4 + w4*x0 + w3*x7 + w3*x6 + w3*x2 + w2*x4 + w1*x6 + w0*x0 + w0]
if not correct_only:
l.append(w7*x6 + w7*x5 + w7*x1 + w6*x7 + w6*x6 + w6*x2 + w5*x7 + w5*x3 + w4*x4 + w3*x5 + w2*x6 + w1*x7 + w0*x0 + 1)
if not biaffine_only:
l.extend([w7**2 + w7*w6 + w7*w3 + w7*w1 + w7*x7 + w7*x6 + w7*x5 + w7*x2 + w7*x1 + w7*x0 + w6**2 + w6*w0 + w6*x6 + w6*x5 + w6*x4 + w6*x3 + w6*x1 + w6*x0 + w5**2 + w5*w4 + w5*w3 \
+ w5*w2 + w5*x7 + w5*x5 + w5*x4 + w5*x1 + w5*x0 + w4**2 + w4*w2 + w4*w0 + w4*x5 + w4*x4 + w4*x2 + w3*w2 + w3*x6 + w3*x3 + w3*x1 + w3*x0 + w2*x7 + w2*x5 + w2*x4 \
+ w2*x0 + w1*x4 + w0**2 + w0*x0,
w7*x6 + w7*x4 + w7*x1 + w6*x7 + w6*x6 + w6*x5 + w6*x2 + w5*x7 + w5*x6 + w5*x5 + w5*x4 + w5*x3 + w5*x1 + w4*x5 + w4*x4 + w4*x3 + w4*x1 + w4*x0 + w3*x7 + w3*x5 + w3*x2 \
+ w2*x7 + w2*x6 + w2*x3 + w1*x7 + w1*x6 + w1*x5 + w1*x4 + w1*x2 + w0*x6 + w0*x5 + w0*x4 + w0*x2 + w0*x1 + x7**2 + x7*x6 + x7*x5 + x7*x3 + x7*x1 + x7*x0 + x6*x2 \
+ x6*x1 + x5*x4 + x5*x3 + x5*x2 + x5*x1 + x4*x3 + x4*x2 + x4*x1 + x3**2 + x3*x2 + x2*x1 + x2*x0,
w7*x5 + w7*x4 + w7*x3 + w7*x1 + w7*x0 + w6*x7 + w6*x5 + w6*x2 + w5*x7 + w5*x6 + w5*x3 + w4*x7 + w4*x6 + w4*x5 + w4*x4 + w4*x2 + w3*x6 + w3*x5 + w3*x4 + w3*x2 + w3*x1 \
+ w2*x6 + w2*x3 + w1*x7 + w1*x4 + w0*x7 + w0*x6 + w0*x5 + w0*x3 + x7*x3 + x7*x2 + x6*x5 + x6*x4 + x6*x3 + x6*x2 + x6*x0 + x5*x4 + x5*x3 + x5*x2 + x4**2 + x4*x3 \
+ x3*x2 + x3*x1,
w7*w3 + w7*w2 + w7*x6 + w7*x5 + w7*x4 + w7*x1 + w7*x0 + w6*w5 + w6*w4 + w6*w3 + w6*w2 + w6*w0 + w6*x5 + w6*x4 + w6*x3 + w6*x2 + w6*x0 + w5*w4 + w5*w3 + w5*w2 + w5*x7 \
+ w5*x6 + w5*x4 + w5*x3 + w5*x0 + w4**2 + w4*w3 + w4*x7 + w4*x4 + w4*x3 + w4*x1 + w3*w2 + w3*w1 + w3*x7 + w3*x5 + w3*x2 + w3*x0 + w2*x6 + w2*x4 + w2*x3 + w1*x7 \
+ w1*x3 + w0*x7,
w7*x5 + w7*x2 + w7*x1 + w6*x7 + w6*x6 + w6*x5 + w6*x4 + w6*x2 + w6*x1 + w5*x5 + w5*x3 + w5*x2 + w4*x3 + w4*x2 + w4*x1 + w3*x6 + w3*x3 + w3*x2 + w3*x0 + w2*x7 + w2*x6 \
+ w2*x5 + w2*x3 + w2*x2 + w1*x6 + w1*x4 + w1*x3 + w0*x4 + w0*x3 + w0*x2 + x7*x5 + x7*x4 + x7*x1 + x7*x0 + x6*x0 + x5**2 + x5*x2 + x5*x1 + x5*x0 + x4**2 + x4*x0 \
+ x3*x2 + x3*x0 + x1**2,
w7*w6 + w7*w5 + w7*w4 + w7*w3 + w7*x7 + w7*x5 + w7*x4 + w7*x3 + w7*x0 + w6**2 + w6*w5 + w6*w4 + w6*w2 + w6*w1 + w6*w0 + w6*x7 + w6*x4 + w6*x3 + w6*x2 + w6*x1 + w5*w4 \
+ w5*w1 + w5*w0 + w5*x7 + w5*x6 + w5*x5 + w5*x3 + w5*x2 + w4*w2 + w4*w1 + w4*x7 + w4*x6 + w4*x3 + w4*x2 + w4*x0 + w3*w0 + w3*x7 + w3*x6 + w3*x4 + w3*x1 + w2**2 \
+ w2*x5 + w2*x3 + w2*x2 + w1*x7 + w1*x6 + w1*x2 + w0*x6,
w7*w5 + w7*w4 + w7*w1 + w7*w0 + w7*x6 + w7*x2 + w6*w0 + w6*x6 + w6*x3 + w6*x2 + w6*x1 + w5**2 + w5*w2 + w5*w1 + w5*w0 + w5*x7 + w5*x6 + w5*x5 + w5*x2 + w4**2 + w4*w0 \
+ w4*x6 + w4*x1 + w4*x0 + w3*w2 + w3*w0 + w3*x5 + w3*x4 + w3*x3 + w3*x2 + w3*x1 + w3*x0 + w2*x7 + w2*x6 + w2*x5 + w2*x4 + w2*x3 + w2*x2 + w2*x0 + w1**2 + w1*x7 \
+ w1*x6 + w1*x4 + w0*x3,
w7*x7 + w7*x6 + w7*x5 + w7*x2 + w6*x7 + w6*x6 + w6*x5 + w6*x4 + w6*x3 + w6*x1 + w5*x5 + w5*x4 + w5*x3 + w5*x1 + w5*x0 + w4*x7 + w4*x5 + w4*x2 + w3*x7 + w3*x6 + w3*x3 \
+ w2*x7 + w2*x6 + w2*x5 + w2*x4 + w2*x2 + w1*x6 + w1*x5 + w1*x4 + w1*x2 + w1*x1 + w0*x6 + w0*x3 + x7**2 + x7*x5 + x7*x3 + x6**2 + x6*x5 + x6*x2 + x6*x0 + x5**2 \
+ x4**2 + x4*x3 + x4*x2 + x4*x1 + x3**2 + x3*x1 + x2*x1,
w7**2 + w7*w6 + w7*w5 + w7*w3 + w7*w1 + w7*w0 + w7*x6 + w7*x5 + w7*x3 + w7*x2 + w7*x1 + w6*w2 + w6*w1 + w6*x7 + w6*x6 + w6*x5 + w6*x2 + w6*x1 + w6*x0 + w5*w4 + w5*w3 \
+ w5*w2 + w5*w1 + w5*x6 + w5*x5 + w5*x4 + w5*x3 + w5*x1 + w5*x0 + w4*w3 + w4*w2 + w4*w1 + w4*x7 + w4*x5 + w4*x4 + w4*x1 + w4*x0 + w3**2 + w3*w2 + w3*x5 + w3*x4 \
+ w3*x2 + w2*w1 + w2*w0 + w2*x6 + w2*x3 + w2*x1 + w2*x0 + w1*x7 + w1*x5 + w1*x4 + w1*x0 + w0*x4,
w7*x7 + w7*x5 + w7*x2 + w6*x7 + w6*x6 + w6*x3 + w5*x7 + w5*x6 + w5*x5 + w5*x4 + w5*x2 + w4*x6 + w4*x5 + w4*x4 + w4*x2 + w4*x1 + w3*x6 + w3*x3 + w2*x7 + w2*x4 + w1*x7 \
+ w1*x6 + w1*x5 + w1*x3 + w0*x7 + w0*x6 + w0*x5 + w0*x3 + w0*x2 + w0*x0 + x7**2 + x7*x6 + x7*x3 + x7*x1 + x6**2 + x6*x0 + x5**2 + x5*x4 + x5*x3 + x5*x2 + x4**2 \
+ x4*x2 + x4*x0 + x3*x2 + x0**2,
w7*x7 + w7*x6 + w7*x5 + w7*x4 + w7*x3 + w7*x1 + w6*x5 + w6*x4 + w6*x3 + w6*x1 + w6*x0 + w5*x7 + w5*x5 + w5*x2 + w4*x7 + w4*x6 + w4*x3 + w3*x7 + w3*x6 + w3*x5 + w3*x4 \
+ w3*x2 + w2*x6 + w2*x5 + w2*x4 + w2*x2 + w2*x1 + w1*x6 + w1*x3 + w0*x7 + w0*x4 + x7*x6 + x7*x5 + x7*x4 + x7*x3 + x6**2 + x6*x5 + x6*x4 + x6*x2 + x6*x1 + x6*x0 \
+ x5*x4 + x5*x1 + x5*x0 + x4*x2 + x4*x1 + x3*x0 + x2**2,
w7*x5 + w7*x4 + w7*x3 + w7*x2 + w6*x7 + w6*x1 + w5*x5 + w5*x4 + w5*x3 + w5*x2 + w5*x1 + w4*x7 + w4*x6 + w4*x4 + w4*x3 + w3*x6 + w3*x5 + w3*x4 + w3*x3 + w2*x2 + w2*x0 \
+ w1*x6 + w1*x5 + w1*x4 + w1*x3 + w1*x2 + w0*x7 + w0*x5 + w0*x4 + x7**2 + x7*x4 + x7*x2 + x6*x4 + x6*x3 + x6*x2 + x6*x1 + x5**2 + x5*x4 + x5*x3 + x5*x2 + x5*x0 \
+ x4*x3 + x4*x2 + x4*x1 + x3**2 + x2*x0 + x1*x0,
w7*x6 + w7*x5 + w7*x3 + w7*x2 + w6*x5 + w6*x4 + w6*x3 + w6*x2 + w5*x7 + w5*x1 + w4*x5 + w4*x4 + w4*x3 + w4*x2 + w4*x1 + w3*x7 + w3*x6 + w3*x4 + w3*x3 + w2*x6 + w2*x5 \
+ w2*x4 + w2*x3 + w1*x2 + w1*x0 + w0*x6 + w0*x5 + w0*x4 + w0*x3 + w0*x2 + x7*x5 + x7*x2 + x7*x0 + x6**2 + x6*x5 + x6*x2 + x6*x1 + x6*x0 + x5**2 + x5*x4 + x4**2 \
+ x4*x2 + x4*x1 + x4*x0 + x3**2 + x3*x2 + x1*x0,
w7**2 + w7*w5 + w7*w3 + w7*x7 + w7*x6 + w7*x4 + w7*x3 + w7*x2 + w6**2 + w6*w5 + w6*w2 + w6*w0 + w6*x7 + w6*x6 + w6*x3 + w6*x2 + w6*x1 + w6*x0 + w5**2 + w5*x7 + w5*x6 \
+ w5*x5 + w5*x4 + w5*x2 + w5*x1 + w4**2 + w4*w3 + w4*w2 + w4*w1 + w4*x6 + w4*x5 + w4*x2 + w4*x1 + w3**2 + w3*w1 + w3*x6 + w3*x5 + w3*x3 + w3*x0 + w2*w1 + w2*x7 \
+ w2*x4 + w2*x2 + w2*x1 + w1*x6 + w1*x5 + w1*x1 + w0*x5,
w7*w5 + w7*w2 + w7*w0 + w7*x5 + w7*x3 + w6**2 + w6*w5 + w6*w2 + w6*w1 + w6*w0 + w6*x7 + w6*x3 + w6*x2 + w6*x0 + w5**2 + w5*w4 + w5*x7 + w5*x6 + w5*x4 + w5*x2 + w5*x0 \
+ w4**2 + w4*w2 + w4*w1 + w4*w0 + w4*x6 + w4*x4 + w4*x3 + w4*x2 + w4*x0 + w3**2 + w3*w2 + w3*x7 + w3*x6 + w3*x4 + w3*x3 + w3*x2 + w3*x0 + w2*x7 + w2*x6 + w2*x4 \
+ w2*x1 + w2*x0 + w1*w0 + w1*x5 + w1*x4 + w0*x1,
w7**2 + w7*w4 + w7*w2 + w7*x6 + w7*x4 + w7*x0 + w6*w4 + w6*w3 + w6*w2 + w6*w1 + w6*x4 + w6*x3 + w6*x1 + w5**2 + w5*w4 + w5*w3 + w5*w2 + w5*w0 + w5*x7 + w5*x5 + w5*x3 \
+ w5*x1 + w5*x0 + w4*w3 + w4*w2 + w4*w1 + w4*x7 + w4*x5 + w4*x4 + w4*x3 + w4*x1 + w4*x0 + w3**2 + w3*x7 + w3*x5 + w3*x4 + w3*x3 + w3*x1 + w2*w0 + w2*x7 + w2*x5 \
+ w2*x2 + w2*x1 + w1*w0 + w1*x6 + w1*x5 + w0*x2])
return l
def _inversion_polynomials_single_sbox(self, x= None, w=None, biaffine_only=None, correct_only=None):
"""
Generate inversion polynomials of a single S-box.
INPUT:
- ``x`` - output variables (default: ``None``)
- ``w`` - input variables (default: ``None``)
- ``biaffine_only`` - only include biaffine polynomials (default: object default)
- ``correct_only`` - only include correct polynomials (default: object default)
EXAMPLES::
sage: sr = mq.SR(1, 1, 1, 8, gf2=True)
sage: len(sr._inversion_polynomials_single_sbox())
24
sage: len(sr._inversion_polynomials_single_sbox(correct_only=True))
23
sage: len(sr._inversion_polynomials_single_sbox(biaffine_only=False))
40
sage: len(sr._inversion_polynomials_single_sbox(biaffine_only=False, correct_only=True))
39
"""
e = self.e
if biaffine_only is None:
biaffine_only = self._biaffine_only
if correct_only is None:
correct_only = self._correct_only
if x is None and w is None:
names = ["w%d" % i for i in reversed(range(e))] + ["x%d"%i for i in reversed(range(e))]
P = PolynomialRing(GF(2), e*2, names, order='lex')
x = Matrix(P, e, 1, P.gens()[e:])
w = Matrix(P, e, 1, P.gens()[:e])
else:
if isinstance(x, (tuple, list)):
P = x[0].parent()
elif is_Matrix(x):
P = x.base_ring()
else:
raise TypeError, "x not understood"
if isinstance(x, (tuple, list)):
x = Matrix(P, e, 1, x)
if isinstance(w, (tuple, list)):
w = Matrix(P, e, 1, w)
T = self._mul_matrix(self.k.gen())
o = Matrix(P, e, 1, [0]*(e-1) + [1])
columns = []
for i in reversed(range(e)):
columns.append((T**i * w).list())
Cw = Matrix(P, e, e, columns).transpose()
columns = []
for i in reversed(range(e)):
columns.append((T**i * x).list())
Cx = Matrix(P, e, e, columns).transpose()
S = self._square_matrix()
l = []
if correct_only:
l.append( (Cw * x + o).list()[:-1] )
else:
l.append( (Cw * x + o).list() )
l.append( (Cw * S *x + x).list() )
l.append( (Cx * S *w + w).list() )
if not biaffine_only:
l.append( ((Cw * S**2 + Cx*S)*x).list() )
l.append( ((Cx * S**2 + Cw*S)*w).list() )
return sum(l, [])
def inversion_polynomials(self, xi, wi, length):
"""
Return polynomials to represent the inversion in the AES S-Box.
INPUT:
- ``xi`` - output variables
- ``wi`` - input variables
- ``length`` - length of both lists
EXAMPLE::
sage: sr = mq.SR(1, 1, 1, 8, gf2=True)
sage: xi = sr.vars('x', 1)
sage: wi = sr.vars('w', 1)
sage: sr.inversion_polynomials(xi, wi, len(xi))[:3]
[x100*w100 + x100*w102 + x100*w103 + x100*w107 + x101*w101 + x101*w102 + x101*w106 + x102*w100 + x102*w101 + x102*w105 + x103*w100 + x103*w104 + x104*w103 + x105*w102 + x106*w101 + x107*w100,
x100*w101 + x100*w103 + x100*w104 + x101*w100 + x101*w102 + x101*w103 + x101*w107 + x102*w101 + x102*w102 + x102*w106 + x103*w100 + x103*w101 + x103*w105 + x104*w100 + x104*w104 + x105*w103 + x106*w102 + x107*w101,
x100*w102 + x100*w104 + x100*w105 + x101*w101 + x101*w103 + x101*w104 + x102*w100 + x102*w102 + x102*w103 + x102*w107 + x103*w101 + x103*w102 + x103*w106 + x104*w100 + x104*w101 + x104*w105 + x105*w100 + x105*w104 + x106*w103 + x107*w102]
"""
if is_Matrix(xi):
xi = xi.list()
if is_Matrix(wi):
wi = wi.list()
e = self.e
l = []
for j in range(0, length, e):
l += self.inversion_polynomials_single_sbox(xi[j:j+e], wi[j:j+e])
return l
def field_polynomials(self, name, i, l=None):
"""
Return list of field polynomials for a given round ``i`` and
name ``name``.
INPUT:
- ``name`` - variable name
- ``i`` - round number
- ``l`` - length of variable list (default: ``None`` = r\*c)
EXAMPLE::
sage: sr = mq.SR(3, 1, 1, 8, gf2=True, polybori=False)
sage: sr.field_polynomials('x', 2)
[x200^2 + x200, x201^2 + x201,
x202^2 + x202, x203^2 + x203,
x204^2 + x204, x205^2 + x205,
x206^2 + x206, x207^2 + x207]
::
sage: sr = mq.SR(3, 1, 1, 8, gf2=True, polybori=True)
sage: sr.field_polynomials('x', 2)
[]
"""
r = self._r
c = self._c
e = self._e
n = self._n
if l is None:
l = r*c
if self._polybori:
return []
_vars = self.vars(name, i, l, e)
return [_vars[e*j+i]**2 - _vars[e*j+i] for j in range(l) for i in range(e)]
class SR_gf2_2(SR_gf2):
"""
This is an example how to customize the SR constructor.
In this example, we replace the S-Box inversion polynomials by the
polynomials generated by the S-Box class.
"""
def inversion_polynomials_single_sbox(self, x=None, w=None, biaffine_only=None, correct_only=None, groebner=False):
"""
Return inversion polynomials of a single S-Box.
INPUT:
- ``x`` - output variables (default: ``None``)
- ``w`` - input variables (default: ``None``)
- ``biaffine_only`` - ignored (always ``False``)
- ``correct_only`` - ignored (always ``True``)
- ``groebner`` - precompute the Groebner basis for this S-Box (default: ``False``).
EXAMPLES::
sage: from sage.crypto.mq.sr import SR_gf2_2
sage: e = 4
sage: sr = SR_gf2_2(1, 1, 1, e)
sage: P = PolynomialRing(GF(2),['x%d'%i for i in range(e)] + ['w%d'%i for i in range(e)],order='lex')
sage: X,W = P.gens()[:e],P.gens()[e:]
sage: sr.inversion_polynomials_single_sbox(X, W, groebner=True)
[x0 + w0*w1*w2 + w0*w1 + w0*w2 + w0*w3 + w0 + w1 + w2,
x1 + w0*w1*w3 + w0*w3 + w0 + w1*w3 + w1 + w2*w3,
x2 + w0*w2*w3 + w0*w2 + w0 + w1*w2 + w1*w3 + w2*w3,
x3 + w0*w1*w2 + w0 + w1*w2*w3 + w1*w2 + w1*w3 + w1 + w2 + w3]
sage: from sage.crypto.mq.sr import SR_gf2_2
sage: e = 4
sage: sr = SR_gf2_2(1, 1, 1, e)
sage: sr.inversion_polynomials_single_sbox()
[w3*w1 + w3*w0 + w3*x2 + w3*x1 + w3 + w2*w1 + w1 + x3 + x2 + x1,
w3*w2 + w3*w1 + w3*x3 + w2 + w1 + x3,
w3*w2 + w3*w1 + w3*x2 + w3 + w2*x3 + x2 + x1,
w3*w2 + w3*w1 + w3*x3 + w3*x2 + w3*x1 + w3 + w2*x2 + w0 + x3 + x2 + x1 + x0,
w3*w2 + w3*w1 + w3*x1 + w3*x0 + w2*x1 + w0 + x3 + x0,
w3*w2 + w3*w1 + w3*w0 + w3*x2 + w3*x1 + w2*w0 + w2*x0 + w0 + x3 + x2 + x1 + x0,
w3*w2 + w3*x1 + w3 + w2*w0 + w1*w0 + w1 + x3 + x2,
w3*w2 + w3*w1 + w3*x1 + w1*x3 + x3 + x2 + x1,
w3*x3 + w3*x2 + w3*x0 + w3 + w1*x2 + w1 + w0 + x2 + x0,
w3*w2 + w3*w1 + w3*x2 + w3*x1 + w1*x1 + w1 + w0 + x2 + x0,
w3*w2 + w3*w1 + w3*w0 + w3*x3 + w3*x1 + w2*w0 + w1*x0 + x3 + x2,
w3*w2 + w3*w1 + w3*x2 + w3*x1 + w3*x0 + w3 + w1 + w0*x3 + x3 + x2,
w3*w2 + w3*w1 + w3*w0 + w3*x3 + w3 + w2*w0 + w1 + w0*x2 + x3 + x2,
w3*w0 + w3*x2 + w2*w0 + w0*x1 + w0 + x3 + x1 + x0,
w3*w0 + w3*x3 + w3*x0 + w2*w0 + w1 + w0*x0 + w0 + x3 + x2,
w3*w2 + w3 + w1 + x3*x2 + x3 + x1,
w3*w2 + w3*x3 + w1 + x3*x1 + x3 + x2,
w3*w2 + w3*w0 + w3*x3 + w3*x2 + w3*x1 + w0 + x3*x0 + x1 + x0,
w3*w2 + w3*w1 + w3*w0 + w3*x3 + w1 + w0 + x2*x1 + x2 + x0,
w3*w2 + w2*w0 + w1 + x3 + x2*x0,
w3*x3 + w3*x1 + w2*w0 + w1 + x3 + x2 + x1*x0 + x1]
TESTS:
Note that ``biaffine_only`` and ``correct_only`` are always
ignored. The former is always false while the second is always
true. They are only accepted for compatibility with the base
class.
sage: from sage.crypto.mq.sr import SR_gf2_2
sage: e = 4
sage: sr = SR_gf2_2(1, 1, 1, e)
sage: l = sr.inversion_polynomials_single_sbox()
sage: l == sr.inversion_polynomials_single_sbox(biaffine_only=True, correct_only=False)
True
"""
e = self.e
if x is None and w is None:
names = ["w%d" % i for i in reversed(range(e))] + ["x%d"%i for i in reversed(range(e))]
P = PolynomialRing(GF(2), e*2, names, order='lex')
x = P.gens()[e:]
w = P.gens()[:e]
S = self.sbox(inversion_only=True)
F = S.polynomials(w, x, degree=e-2, groebner=groebner)
return F
class AllowZeroInversionsContext:
"""
Temporarily allow zero inversion.
"""
def __init__(self, sr):
"""
EXAMPLE::
sage: from sage.crypto.mq.sr import AllowZeroInversionsContext
sage: sr = mq.SR(1,2,2,4)
sage: with AllowZeroInversionsContext(sr):
... sr.sub_byte(0)
a^2 + a
"""
self.sr = sr
def __enter__(self):
"""
EXAMPLE::
sage: from sage.crypto.mq.sr import AllowZeroInversionsContext
sage: sr = mq.SR(1,2,2,4)
sage: sr.sub_byte(0)
Traceback (most recent call last):
...
ZeroDivisionError: A zero inversion occurred during an encryption or key schedule.
sage: with AllowZeroInversionsContext(sr):
... sr.sub_byte(0)
a^2 + a
"""
self.allow_zero_inversions = self.sr._allow_zero_inversions
self.sr._allow_zero_inversions = True
def __exit__(self, typ, value, tb):
"""
EXAMPLE::
sage: from sage.crypto.mq.sr import AllowZeroInversionsContext
sage: sr = mq.SR(1,2,2,4)
sage: with AllowZeroInversionsContext(sr):
... sr.sub_byte(0)
a^2 + a
sage: sr._allow_zero_inversions
False
"""
self.sr._allow_zero_inversions = self.allow_zero_inversions
def test_consistency(max_n=2, **kwargs):
r"""
Test all combinations of ``r``, ``c``, ``e`` and ``n`` in ``(1,
2)`` for consistency of random encryptions and their polynomial
systems. `\GF{2}` and `\GF{2^e}` systems are tested. This test takes
a while.
INPUT:
- ``max_n`` -- maximal number of rounds to consider (default: 2)
- ``kwargs`` -- are passed to the SR constructor
TESTS:
The following test called with ``max_n`` = 2 requires a LOT of RAM
(much more than 2GB). Since this might cause the doctest to fail
on machines with "only" 2GB of RAM, we test ``max_n`` = 1, which
has a more reasonable memory usage. ::
sage: from sage.crypto.mq.sr import test_consistency
sage: test_consistency(1) # long time (65s on sage.math, 2012)
True
"""
consistent = True
for r in (1, 2, 4):
for c in (1, 2, 4):
for e in (4, 8):
for n in range(1, max_n+1):
for gf2 in (True, False):
zero_division = True
while zero_division:
sr = SR(n, r, c, e, gf2=gf2, **kwargs)
try:
F, s = sr.polynomial_system()
F = F.subs(s)
consistent &= (F.groebner_basis()[0] != 1)
if not consistent:
print sr, " is not consistent"
zero_division = False
except ZeroDivisionError:
pass
return consistent
