A family of binary group operations for block cipher applications

A family of multiplication-like group operations over m-bit words is investigated. The operations are defined by a homomorphic mapping of binary words into the elements of a multiplicative group modulo 2^kF_n, where k is an integer ⩾1, and F_n is the n -th Fermat prime (n=0,1,2,3,4). The order of the multiplicative group is 2^N+k-1, where N=2ⁿ. The operations are usable for any given value of m if k and n are chosen such that m =2ⁿ+k-1. Expressions are presented for the mapping function for each value of n, and their inverses. Application of the results in block cipher design is discussed