Power Pell Sequences, Some Periodic Relations of These Sequences, and a Cryptographic Application With Power Pell Sequences

Abstract

Here, first of all, the authors investigated power Fibonacci sequence modulo k and formulas for the periods of these sequences, based on the period of the Fibonacci sequence modulo k. And then, the authors described a new power sequence for positive integer modulus. They named these sequences power Pell sequences modulo k. After that the authors determined those positive integer moduli for which this sequence exists and the number of such sequences for a given modulo k. In addition, the authors provide formulas for the periods of these sequences, based on the period of the Pell sequence modulo k, and they studied sequence/subsequence relationships between power Pell sequences. Finally, the authors examined ElGamal cryptosystem which is one of the asymmetric cryptographic systems and ElGamal cryptosystem which is obtained by using some power sequences. And they obtained asymmetric cryptographic applications by using power Pell sequences which the authors described.