Relationship between Fibonacci and Lucas sequences and their application in Symmetric Cryptosystems

Abstract— Identification whether a certain number belongs to either Fibonacci or Lucas sequence is computationally intense task in the sense that the sequence itself need to be written from the beginning until the requested number. In this paper we present a method of detecting whether a certain number is of Lucas or Fibonacci sequence as well as introduce relationships between such sequences. This relationship is being used for creation of encryption/decryption methods by utilizing symmetrical key generated by elements of both Lucas and Fibonacci sequences. The strength of this symmetrical cryptosystem lies in the introduced mathematical operators called pentors and ultra pentors which are hard to generate and guess by brute force computation.