Polynomial Basis

# 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

[itex]
[/itex]

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

[itex]
[/itex]

using Zech's logarithms.

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

[itex](2alpha^2 + 2alpha + 1) + (2alpha + 2) = 2alpha^2 + 4alpha + 3 mod = 2alpha^2 + alpha[/itex]

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> &minus; m additions in GF(p).

Some of the methods for reducing these values include:

• Lookup tables &mdash; a prestored table of results; mainly used for small fields, otherwise the table......
• ...

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