A novel public key cryptography based on generalized Lucas matrices

Abstract

In this article, we have proposed a generalized Lucas matrix (recursive matrix of higher order) having relation with generalized Fibonacci sequences and established many special properties in addition to that usual matrix algebra. Further, we have proposed a modified public key cryptography using these matrices as keys in Affine cipher and key agreement for encryption-decryption with the combination of terms of generalized Lucas sequences under residue operations. In this scheme, instead of exchanging the whole key matrix, only a pair of numbers(parameters) need to be exchanged, which reduces the time complexity as well as space complexity of the key transmission and has a large key-space.

Cryptography utilizing the Affine-Hill cipher and Extended Generalized Fibonacci matrices

“…Let us consider alphabets defined as follows for letters from A-Z equivalent to 00-25, digits 0-9 are that to 26-35 and 36 for blank space/white space. Therefore, the numerical values equivalent to "SUMAN2022" is [18,20,12,00,13,28,26,28,28]. Now according to the algorithm (4.1), Alice first choose an integer ω such that 1 < ω < ϕ(37), say ω = 22.…”