Linear codes and ciphers¶
Codes¶
A linear code of length is a finite dimensional
subspace of
. Sage can compute with linear
error-correcting codes to a limited extent. It basically has some
wrappers to GAP and GUAVA commands. GUAVA 2.8 is not included
with Sage 4.0’s install of GAP but can be installed as an optional
package.
Sage can compute Hamming codes
sage: C = codes.HammingCode(GF(3), 3)
sage: C
[13, 10] Hamming Code over GF(3)
sage: C.minimum_distance()
3
sage: C.generator_matrix()
[1 0 0 0 0 0 0 0 0 0 1 2 0]
[0 1 0 0 0 0 0 0 0 0 0 1 2]
[0 0 1 0 0 0 0 0 0 0 1 0 2]
[0 0 0 1 0 0 0 0 0 0 1 1 1]
[0 0 0 0 1 0 0 0 0 0 1 1 2]
[0 0 0 0 0 1 0 0 0 0 2 0 2]
[0 0 0 0 0 0 1 0 0 0 1 2 1]
[0 0 0 0 0 0 0 1 0 0 2 1 1]
[0 0 0 0 0 0 0 0 1 0 2 2 0]
[0 0 0 0 0 0 0 0 0 1 0 1 1]
the four Golay codes
sage: C = codes.GolayCode(GF(3))
sage: C
[12, 6, 6] Extended Golay code over Finite Field of size 3
sage: C.minimum_distance()
6
sage: C.generator_matrix()
[1 0 0 0 0 0 2 0 1 2 1 2]
[0 1 0 0 0 0 1 2 2 2 1 0]
[0 0 1 0 0 0 1 1 1 0 1 1]
[0 0 0 1 0 0 1 1 0 2 2 2]
[0 0 0 0 1 0 2 1 2 2 0 1]
[0 0 0 0 0 1 0 2 1 2 2 1]
as well as binary Reed-Muller codes, quadratic residue codes, quasi-quadratic residue codes, “random” linear codes, and a code generated by a matrix of full rank (using, as usual, the rows as the basis).
For a given code, , Sage can return a generator matrix,
a check matrix, and the dual code:
sage: C = codes.HammingCode(GF(2), 3)
sage: Cperp = C.dual_code()
sage: C; Cperp
[7, 4] Hamming Code over GF(2)
[7, 3] linear code over GF(2)
sage: C.generator_matrix()
[1 0 0 0 0 1 1]
[0 1 0 0 1 0 1]
[0 0 1 0 1 1 0]
[0 0 0 1 1 1 1]
sage: C.parity_check_matrix()
[1 0 1 0 1 0 1]
[0 1 1 0 0 1 1]
[0 0 0 1 1 1 1]
sage: C.dual_code()
[7, 3] linear code over GF(2)
sage: C = codes.HammingCode(GF(4,'a'), 3)
sage: C.dual_code()
[21, 3] linear code over GF(4)
For and a vector
, Sage can try
to decode
(i.e., find the codeword
closest to
in the Hamming metric) using syndrome
decoding. As of yet, no special decoding methods have been
implemented.
sage: C = codes.HammingCode(GF(2), 3)
sage: MS = MatrixSpace(GF(2),1,7)
sage: F = GF(2); a = F.gen()
sage: v = vector([a,a,F(0),a,a,F(0),a])
sage: c = C.decode_to_code(v, "Syndrome"); c
(1, 1, 0, 1, 0, 0, 1)
sage: c in C
True
To plot the (histogram of) the weight distribution of a code, one can use the matplotlib package included with Sage:
sage: C = codes.HammingCode(GF(2), 4)
sage: C
[15, 11] Hamming Code over GF(2)
sage: w = C.weight_distribution(); w
[1, 0, 0, 35, 105, 168, 280, 435, 435, 280, 168, 105, 35, 0, 0, 1]
sage: J = range(len(w))
sage: W = IndexedSequence([ZZ(w[i]) for i in J],J)
sage: P = W.plot_histogram()
Now type show(P)
to view this.
There are several coding theory functions we are skipping entirely.
Please see the reference manual or the file
coding/linear_codes.py
for examples.
Sage can also compute algebraic-geometric codes, called AG codes,
via the Singular interface § sec:agcodes. One may also use the AG
codes implemented in GUAVA via the Sage interface to GAP
gap_console()
. See the GUAVA manual for more details. {GUAVA}
Ciphers¶
LFSRs¶
A special type of stream cipher is implemented in Sage, namely, a linear feedback shift register (LFSR) sequence defined over a finite field. Stream ciphers have been used for a long time as a source of pseudo-random number generators. {linear feedback shift register}
S. Golomb {G} gives a list of three statistical properties a
sequence of numbers ,
, should display to be considered “random”.
Define the autocorrelation of
to be
In the case where is periodic with period
then this reduces to
Assume is periodic with period
.
balance:
.
low autocorrelation:
(For sequences satisfying these first two properties, it is known that
must hold.)
proportional runs property: In each period, half the runs have length
, one-fourth have length
, etc. Moveover, there are as many runs of
‘s as there are of
‘s.
A sequence satisfying these properties will be called pseudo-random. {pseudo-random}
A general feedback shift register is a map
of the form
where is a given
function. When
is of the form
..math:: C(x_0,...,x_{n-1})=c_0x_0+...+c_{n-1}x_{n-1},
for some given constants , the map is
called a linear feedback shift register (LFSR). The sequence of
coefficients
is called the key and the polynomial
is sometimes called the connection polynomial.
Example: Over , if
then
,
The LFSR sequence is then
The sequence of ‘s is periodic with period
and satisfies Golomb’s three randomness
conditions. However, this sequence of period 15 can be “cracked”
(i.e., a procedure to reproduce
) by knowing only 8
terms! This is the function of the Berlekamp-Massey algorithm {M},
implemented as
lfsr_connection_polynomial
(which produces the
reverse of berlekamp_massey
).
sage: F = GF(2)
sage: o = F(0)
sage: l = F(1)
sage: key = [l,o,o,l]; fill = [l,l,o,l]; n = 20
sage: s = lfsr_sequence(key,fill,n); s
[1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0]
sage: lfsr_autocorrelation(s,15,7)
4/15
sage: lfsr_autocorrelation(s,15,0)
8/15
sage: lfsr_connection_polynomial(s)
x^4 + x + 1
sage: berlekamp_massey(s)
x^4 + x^3 + 1
Classical ciphers¶
has a type for cryptosystems (created by David Kohel, who also wrote the examples below), implementing classical cryptosystems. The general interface is as follows:
sage: S = AlphabeticStrings()
sage: S
Free alphabetic string monoid on A-Z
sage: E = SubstitutionCryptosystem(S)
sage: E
Substitution cryptosystem on Free alphabetic string monoid on A-Z
sage: K = S([ 25-i for i in range(26) ])
sage: e = E(K)
sage: m = S("THECATINTHEHAT")
sage: e(m)
GSVXZGRMGSVSZG
Here’s another example:
sage: S = AlphabeticStrings()
sage: E = TranspositionCryptosystem(S,15);
sage: m = S("THECATANDTHEHAT")
sage: G = E.key_space()
sage: G
Symmetric group of order 15! as a permutation group
sage: g = G([ 3, 2, 1, 6, 5, 4, 9, 8, 7, 12, 11, 10, 15, 14, 13 ])
sage: e = E(g)
sage: e(m)
EHTTACDNAEHTTAH
The idea is that a cryptosystem is a map
where
,
, and
are the key space,
plaintext (or message) space, and ciphertext space, respectively.
is presumed to be injective, so
e.key()
returns the
pre-image key.