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

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

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 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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