Topics from finite fields

Abstract

Throughout this chapter we use the following notation: p denotes a prime number, n a natural number, q:= p ⁿ. F = F _q is the finite field of p ⁿ elements and w a primitive root of F, i.e. F ^× = 〈w〉. In general, arithmetic in F will be done by using two representations for its elements x: (i)

$$x = \sum\nolimits_{{i = 1}}^{n} {{{x}_{i}}{{w}_{i}}({{x}_{i}}\epsilon {{F}_{p}};{{w}_{n}},an{{F}_{p}} - basisofF)}$$

, (ii)

$$x = {{w}^{v}}\left( {v \in \left\{ {0,1, \ldots ,q - 2} \right\}forx \ne 0,v = \infty forx = 0} \right)$$

. Then addition and subtraction is done by the first, multiplication and division by the second representation. Thus all we need are two tables allowing to switch from one representation to the other. These ideas are useful if p ⁿ is not really large.