The Fifth and the Sixth Order Gopala Hemachandra Representations and the Use of These Representations in Symmetric Cryptography

Abstract

We all know that every positive integer has a unique Fibonacci representation, but some positive integers have multiple Gopala Hemachandra (GH) representations, or some positive integers haven't any GH representation. Here, the authors found the first k-positive integer k=(3 2^((m-1))-1) for which there is no Zeckendorf's representation for Gopala Hemachandra sequence whose order m. Thus, the authors formulated the first positive integer whose Zeckendorf's representation can't be found in terms of its order. The authors also described the fourth, the fifth, and the sixth order GH representation of positive integers and obtained the fifth and the sixth order GH representations of the first 26 positive integers uniformly according to a certain rule with a table. Finally, the authors used these GH representations in symmetric cryptography, and the authors made some applications with a method which they construct similar to Nalli and Ozyilmaz.