C. Koscielny
Computing in GF(p^m) and in gff(n^m) using Maple

It is mentioned in the previous volume that the author intends to show how to construct strong ciphers using SMG(p^m). But in order to implement such cryptosystems, an effective tool for computing in GF(p^m) and SMG(p^m), in the form of an appropriate hardware or software, is needed. The operation of this hardware or software ought to be defined by means of the detailed algorithms. Thus, to get ready the execution of his intention, the author describes in the paper these algorithms, which are represented as routines, written in the comprehensive Maple interpreter, intelligible both for mathematicians and programmers as well. The routines may be used either immediately as elements of encrypting/decrypting procedures in the Maple programming environment or can be easily translated into any compiled programming language (in this case encryption/decryption can be performed at least 100 times faster than in the Maple environment). Aside from that on the basis of the mentioned routines any VLSI chip as the encrypting/decrypting hardware for SMG(p^m)- and GF(p^m)-based cryptosystems can be produced.

It has also been shown that SMG(p^m) can be considered as a multiplicative system of an algebraic structure with addition and multiplication operations, containing a large class of systems, including GF(p^m). The system is denoted as gff(n^m), and multiplication in it is performed modulo an arbitrary polynomial od degree $m$ over the ring Z_n. That way gff(n^m) is a generalization of Galois field, very well suited for applications in cryptography. This system is named a generalized finite field.