Using offset invariant crossover as a tool for discovering cycle lengths of a periodic function

Abstract

We investigate the problem of discovering a multiple of the cycle length of a periodic function. A generalization of the crossover operator is proposed which is offset invariant, i.e. it functions in an unchanged manner if a constant offset is added to both parents. The properties of this operator are investigated using a mixture of theoretical and empirical techniques. Its convergence is demonstrated even in the extreme case of a near random evaluation function. We investigate its application to the use of optimization to find some multiple of the period of a periodic function. This investigation takes place in the context of an exponential function defined over the integers modulo a composite number. Competing techniques to accomplish this using optimization are demonstrated and their growth rates empirically investigated. A comparison of the performance of normal crossover and offset invariant crossover demonstrates an advantage for the offset invariant crossover operator. Finally we show how our methods can be applied to problems of RSA cryptography.