Introduction to Cryptography with Coding Theory, 3rd Edition

B.5 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 $0\text{,}1\text{,}0\text{,}0\text{,}0\text{.}$

SOLUTION

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

> lfsr([1, 0, 1, 0, 0], [0, 1, 0, 0, 0], 50)
[0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 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 recurrence that generates this sequence.

SOLUTION

First, we need to find the length of the recurrence. The command lfsrlength(v, n) calculates the determinants ...