Introduction to Cryptography with Coding Theory, 3rd Edition

C.4 Examples for Chapter 5

Example 18

Compute the first 50 terms of the recurrence

$$x_{n+5}\equiv x_n+x_{n+2}(\text{mod}2)\text{.}$$

The initial values are $0\text{,}1\text{,}0\text{,}0\text{,}0$.

SOLUTION

The vector of coefficients is $[1\text{,}0\text{,}1\text{,}0\text{,}0]$ and the initial values are given by the vector $[0\text{,}1\text{,}0\text{,}0\text{,}0]$. Type

>> lfsr([1 0 1 0 0],[0 1 0 0 0],50)

ans =
   Columns 1 through 12
    0   1   0   0   0   0   1   0   0   1   0   1
   Columns 13 through 24
    1   0   0   1   1   1   1   1   0   0   0   1
   Columns 25 through 36
    1   0   1   1   1   0   1   0   1   0   0   0
   Columns 37 through 48
    0   1   0   0   1   0   1   1   0   0   1   1
   Columns 49 through 50
    1   1

Example 19

Suppose the first 20 terms of an LFSR sequence are 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1. Find a recursion that generates this sequence.

SOLUTION

First, we find a candidate for the length ...