Circuits and methods for performing exponentiation and inversion of finite field elements

摘要

An exponentiation circuit for computing an exponential power of a finite field element includes combinatory logic circuits that map input digits of a multi-digit field element β to output digits of an output multi-digit field element β2m. The exponentiation circuit is capable of computing a power of a field element without performing any multiplication operations and requires only exclusive-OR logic operations to generate the output exponential field element. A circuit for generating a multiplicative inverse of a finite field element can be constructed from a set of parallel exponentiation circuits, with each of the parallel exponentiation circuits generating a different multi-digit field element β2m directly from the input field element β. Multiplier circuits multiply together the outputs of the parallel exponentiation circuits to generate the multiplicative inverse of the field element β.