Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 4H (Coding and Cryptography)

A general feedback shift register of length $d$ over an alphabet $A$ stores $(s_n,\dots ,s_{n+d-1})$, outputs $s_n$, and shifts to $(s_{n+1},\dots ,s_{n+d-1},F(s_n,\dots ,s_{n+d-1}))$ for a fixed function $F:A^d\to A$. When the alphabet is a field $K$, it is a linear-feedback shift register if $F$ is linear over $K$, equivalently $s_{n+d}={\sum }_{j=0}^{d-1}{a}_j{s}_{n+j}$ for fixed coefficients in $K$. The binary case uses the finite field $K={\mathbb{F}}_2$; the examples and counterexamples below are binary.

For a genuinely nonlinear example take length three and $F(a,b,c)=a+bc$. Its values at $(0,1,0)$ and $(0,0,1)$ are both zero, but its value at their sum $(0,1,1)$ is one, violating additivity. Thus this general register is not linear. It even has an invertible state update: from $(b,c,d)$ the old leading bit is $a=d+bc$, so the counterexample does not rely on allowing singular registers.

False. Take $s_{n+1}=s_n$ and $t_{n+2}=t_{n+1}+t_n$. The only nonzero length-one fill produces constant ones. Constant ones cannot satisfy the second linear recurrence relation, since $1+1=0$ in ${\mathbb{F}}_2$. Thus these linear-feedback shift registers have no identical outputs from nonzero fills.

False. Take $s_{n+1}=s_n$ and $t_{n+2}=t_n$. Fills $1$ and $(1,1)$ both produce the constant-one sequence, despite different lengths. Both state updates are invertible.

True. A length-$d$ linear-feedback shift register has $2^d-1$ nonzero fills, and its first $d$ output bits recover its fill, so these fills give distinct output sequences. If lengths satisfy $r<s$, the longer register has strictly more nonzero outputs. At least one is absent from the shorter register's output set; choosing any nonzero shorter fill then gives unequal outputs. This is the counting property of output sets of different-length linear-feedback registers.

True. The length-one and length-two registers in part (ii), with the displayed nonzero fills, provide an explicit example. Register length need not equal the minimal recurrence length of every sequence it can generate.