Polynomial Basis

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In mathematics, the **polynomial basis** is a basis for finite extensions of finite fields.

Let α ∈ GF(*p*<sup>*m*</sup>) be the root of a primitive polynomial of degree *m* over GF(*p*). The polynomial basis of GF(*p*<sup>*m*</sup>) is then

The set of elements of GF(*p*<sup>*m*</sup>) can then be represented as:

using Zech's logarithms.

## Addition

Addition using the polynomial basis is as simple as addition modulo *p*. For example, in GF(3<sup>*m*</sup>):

In GF(2<sup>*m*</sup>), addition is especially easy, since addition and subtraction modulo 2 are the same thing, and furthermore this operation can be done in hardware using the basic XOR logic gate.

## Multiplication

Multiplication of two elements in the polynomial basis can be accomplished in the normal way of multiplication, but there are a number of ways to speed up multiplication, especially in hardware. Using the straightforward method to multiply two elements in GF(*p*<sup>*m*</sup>) requires up to *m*<sup>2</sup> multiplications in GF(*p*) and up to *m*<sup>2</sup> − *m* additions in GF(*p*).

Some of the methods for reducing these values include:

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Let α ∈ GF(

- <math>

The set of elements of GF(

- <math>

using Zech's logarithms.

- <math>(2alpha^2 + 2alpha + 1) + (2alpha + 2) = 2alpha^2 + 4alpha + 3 mod = 2alpha^2 + alpha</math>

In GF(2<sup>

Some of the methods for reducing these values include:

- Lookup tables — a prestored table of results; mainly used for small fields, otherwise the table...... ...

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