Implementation of log and inverse operation in a Galois Field

摘要

A method, including factoring an order of a multiplicative group of a Galois Field to produce a first integer factor p and a second integer factor q, wherein the multiplicative group includes (2m−1) elements, m a non-negative integer, so that 2m−1=pq. The method further includes receiving an element x of the Galois Field expressible as α(qi+j), where α is a primitive element of the group, i is a first non-negative integer less than p, and j is a second integer less than q. An inverse or a logarithm of the element x is calculated as a function of qi and j.

主权项

1. A method, comprising:
providing a factoring of an order of a multiplicative group of a Galois Field including 2m−1 elements, and having a primitive element α, into a first integer factor p and a second integer factor q, wherein p and q are both greater than 1, so that 2m−1=pq; receiving, by at least one processor, an element x of the Galois Field; determining, by the at least one processor, a first value j equal to e mod q, where e is an integer such that αe=x; determining, by the at least one processor, a second value which can be used together with the first value to determine a logarithm of the received element x; determining, by the at least one processor, a logarithm of the element x as a function of the first and second values; and encrypting, decrypting or decoding a code responsive to the determined logarithm.