# Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 4 Solutions - Ex4.8

- 1 min

## Exercise 4.8

Find all irreducible polynomials

1. of degree 3 over $GF(2)$,
2. of degree 4 over $GF(2)$.

The best approach for doing this is to consider all polynomials of lower degree and check whether they are factors. Please note that we only consider monic irreducible polynomials, i.e., polynomials with the highest coefficient equal to one.

### Solution

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1. There are two irreducible polynomials in $GF(2)$ of degree 3:

•   $x^3+x+1$

•   $x^3+x^2+1$

2. There are three irreducible polynomials in $GF(2)$ of degree 4:

•   $x^4+x+1$

•   $x^4+x^3+x^2+x+1$

•   $x^4+x^3+1$