Linear codes and ciphers¶

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 \({\bf a}=\{a_n\}_{n=1}^\infty\), \(a_n\in \{0,1\}\), should display to be considered “random”. Define the autocorrelation of \({\bf a}\) to be

\[C(k)=C(k,{\bf a})=\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{n=1}^N (-1)^{a_n+a_{n+k}}.\]

In the case where \(a\) is periodic with period \(P\) then this reduces to

Assume \(a\) is periodic with period \(P\).

balance: \(|\sum_{n=1}^P(-1)^{a_n}|\leq 1\).
low autocorrelation:

\[\begin{split}C(k)= \left\{ \begin{array}{cc} 1,& k=0,\\ \epsilon, & k\not= 0. \end{array} \right.\end{split}\]

(For sequences satisfying these first two properties, it is known that \(\epsilon=-1/P\) must hold.)
proportional runs property: In each period, half the runs have length \(1\), one-fourth have length \(2\), etc. Moveover, there are as many runs of \(1\)’s as there are of \(0\)’s.

A sequence satisfying these properties will be called pseudo-random. {pseudo-random}

A general feedback shift register is a map \(f:{\bf F}_q^d\rightarrow {\bf F}_q^d\) of the form

\[\begin{split}\begin{array}{c} f(x_0,...,x_{n-1})=(x_1,x_2,...,x_n),\\ x_n=C(x_0,...,x_{n-1}), \end{array}\end{split}\]

where \(C:{\bf F}_q^d\rightarrow {\bf F}_q\) is a given function. When \(C\) is of the form

\[C(x_0,...,x_{n-1}) = c_0 x_0 + ... + c_{n-1} x_{n-1},\]

for some given constants \(c_i\in {\bf F}_q\), the map is called a linear feedback shift register (LFSR). The sequence of coefficients \(c_i\) is called the key and the polynomial

\[C(x) = 1+ c_0x +...+c_{n-1}x^n\]

is sometimes called the connection polynomial.

Example: Over \(GF(2)\), if \([c_0,c_1,c_2,c_3]=[1,0,0,1]\) then \(C(x) = 1 + x + x^4\),

The LFSR sequence is then

\[\begin{split}\begin{array}{c} 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, \\ 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, ...\ . \end{array}\end{split}\]

The sequence of \(0,1\)’s is periodic with period \(P=2^4-1=15\) and satisfies Golomb’s three randomness conditions. However, this sequence of period 15 can be “cracked” (i.e., a procedure to reproduce \(g(x)\)) 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

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: from sage.matrix.berlekamp_massey import berlekamp_massey
sage: berlekamp_massey(s)
x^4 + x^3 + 1

Python

>>> from sage.all import *
>>> F = GF(Integer(2))
>>> o = F(Integer(0))
>>> l = F(Integer(1))
>>> key = [l,o,o,l]; fill = [l,l,o,l]; n = Integer(20)
>>> 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]
>>> lfsr_autocorrelation(s,Integer(15),Integer(7))
4/15
>>> lfsr_autocorrelation(s,Integer(15),Integer(0))
8/15
>>> lfsr_connection_polynomial(s)
x^4 + x + 1
>>> from sage.matrix.berlekamp_massey import berlekamp_massey
>>> 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

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

Python

>>> from sage.all import *
>>> S = AlphabeticStrings()
>>> S
Free alphabetic string monoid on A-Z
>>> E = SubstitutionCryptosystem(S)
>>> E
Substitution cryptosystem on Free alphabetic string monoid on A-Z
>>> K = S([ Integer(25)-i for i in range(Integer(26)) ])
>>> e = E(K)
>>> m = S("THECATINTHEHAT")
>>> e(m)
GSVXZGRMGSVSZG

Here’s another example:

Sage

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

Python

>>> from sage.all import *
>>> S = AlphabeticStrings()
>>> E = TranspositionCryptosystem(S,Integer(15));
>>> m = S("THECATANDTHEHAT")
>>> G = E.key_space()
>>> G
Symmetric group of order 15! as a permutation group
>>> g = G([ Integer(3), Integer(2), Integer(1), Integer(6), Integer(5), Integer(4), Integer(9), Integer(8), Integer(7), Integer(12), Integer(11), Integer(10), Integer(15), Integer(14), Integer(13) ])
>>> e = E(g)
>>> e(m)
EHTTACDNAEHTTAH

The idea is that a cryptosystem is a map \(E: KS \to \text{Hom}_\text{Set}(MS,CS)\) where \(KS\), \(MS\), and \(CS\) are the key space, plaintext (or message) space, and ciphertext space, respectively. \(E\) is presumed to be injective, so e.key() returns the pre-image key.

Linear codes and ciphers¶

Codes¶

Ciphers¶

LFSRs¶

Classical ciphers¶