Finite field inversion over the dual basis

Abstract
In this transaction brief we consider the design of dual basis inversion circuits for GF(2/sup m/). Two architectures are presented-one bit-serial and one bit-parallel-both of which are based on Fermat's theorem. Finite field inverters based on Fermat's theorem have previously been presented which operate over the normal basis and the polynomial basis. However there are two advantages to be gained by forcing inversion circuits to operate over the dual basis. First, these inversion circuits can be utilized in circuits using hardware efficient dual basis multipliers without any extra basis converters. And second, the inversion circuits themselves can take advantage of dual basis multipliers, thus reducing their own hardware levels. As both these approaches require squaring in a finite field to take place, a theorem is presented which allows circuits to be easily designed to carry out squaring over the dual basis.

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