# Cryptography

## Linear feedback shift registers (LFSRs)

A special type of stream cipher is implemented in Sage, namely, a LFSR sequence defined over a finite field. Stream ciphers have been used for a long time as a source of pseudo-random number generators.

S. Golomb ("Shift register sequences", Aegean Park Press, Laguna Hills, Ca, 1967) gives a list of three statistical properties a sequence of numbers \mathbf{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

In the case where {\bf a} is periodic with period P then this reduces to

Assume {\bf a} is periodic with period P .

*balance*: \vert\sum_{n=1}^P(-1)^{a_n}\vert\leq 1 .*low autocorrelation*:- C(k)= \left\{ \begin{array}{cc} 1,& k=0,\\ \epsilon, & k\not= 0. \end{array}\right.
(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. Moreover, there are as many runs of 1 's as there are of 0 's.

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

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

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

\displaystyle C(x_0,...,x_{n-1})=c_0x_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

\displaystyle 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 ,

\displaystyle x_n = x_{n-4} + x_{n-1}, n\geq 4.

The LFSR sequence is then

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 (James L. Massey, Shift-Register Synthesis and BCH Decoding, IEEE Trans. on Information Theory, vol. 15(1), pp. 122-127, Jan 1969.), 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 crytosystems

Sage 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 E: KS \rightarrow Hom_{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.