GaloisCounter

GaloisCounter is a class of counter generators based on arithmetic in a Galois field, intended for deterministic sequencing in hardware and software. In the common design, the current state is an element of GF(2^m); a fixed primitive polynomial defines the field, and the next state is obtained by multiplying by a fixed primitive element α in GF(2^m). If the state is nonzero, repeating this multiplication traverses all 2^m − 1 nonzero elements before cycling back; the zero state is typically excluded to maintain a maximal period.

This structure allows efficient hardware implementation using simple feedback logic or an LFSR-like circuit. The period is maximal at 2^m − 1, and the output distribution over states is uniform among nonzero elements. The sequence is linear over GF(2), enabling fast jump-ahead by exponentiation, which is useful for synchronized streams or reproducible test patterns.

Variants exist that add stepwise or alternate representations, such as using GF(p^n) arithmetic with a primitive element, or combining multiplicative and additive steps to adjust the cycle or distribution.

Applications include systematic test-pattern generation, scramblers, and certain PRNG tasks where a long, repeatable cycle and predictable statistics are desirable. For cryptographic purposes, GaloisCounter designs are generally not considered secure by themselves and should be used only within established cryptographic protocols or replaced by validated cryptographic primitives.