# Modular reduction in Galois fields

I want to compute x^6 mod x^5+x^2+1 in the Galois Field GF(2^5). Does anyone know how to do this in SAGE?

Modular reduction in Galois fields

add a comment

3

First, define `k`

to be the field GF(2^5), whose generator is named `a`

:

```
sage: k.<a> = FiniteField(2^5); k
Finite Field in a of size 2^5
```

Alternatively :

```
sage: k = GF(2^5, 'a'); k
Finite Field in a of size 2^5
```

Then, define the polynomial ring `k[x]`

:

```
sage: R.<x> = PolynomialRing(k); R
Univariate Polynomial Ring in x over Finite Field in a of size 2^5
```

Alternatively:

```
sage: R = k['x']; R
Univariate Polynomial Ring in x over Finite Field in a of size 2^5
```

Then, do your computation in R:

```
sage: P = R(x^6)
sage: P.mod(x^5+x^2+1)
x^3 + x
```

Asked: **
2013-04-28 00:04:04 -0500
**

Seen: **478 times**

Last updated: **Apr 28 '13**

Finding order of a polynomial over finite field

Checking Similarity of Matrices over Finite Fields

What is the effect of declaring a polynomial `sparse'?

Order of randomly generated elliptic curve

Defining Polynomial Basis and Generic Polynomials

Create Morphism's between Finite Fields and VectorSpaces

Solving modular systems of equation

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.