Cryptographical Applications of Linear Transformations and Pell Equations: An R-Based Approach

Objective: This article’s main aim is to encrypt and decrypt the message to be sent using matrices of the linear transformation and the fundamental solution of the Pell equation. R Programming is provided at the end of the article. Methods: The encryption and decryption algorithms were created using the fundamental solutions of the Pell equation 𝑥2 − 𝑝𝑦2 = 1, where p is a positive integer which is not a perfect square, and using the matrices of the linear transformations. The message to be sent is converted into block matrices of order 2. Based on the number of blocks, the prime 𝑝 is determined. Findings: Here, we gave two examples: manually for the sentence ”MATHS IS A UNIVERSAL SUBJECT” and using R programming for the word ”PIE.” Novelty: The concept of encrypting and decrypting a message or word already exists. Many algorithms have been created in different ways. But, in this article, we created the algorithms connecting the matrix of linear transformations and the fundamental solution of Pell equations. 2020 MSC Classification: 94A60, 15A04, 11C20. Keywords: Cryptography; Linear transformation; Pell equation; Encryption; Decryption