Encryption Algorithms with Linear Algebra and Number Theory

Abstract

Although there are highly sophisticated encryption schemes that have been developed in the past, such as Elliptical Curve Encryption, Secure-Hash-Algorithm and Data Encryption Standard, there is still much to explore in cryptography. The goal of this research is to explore encryption techniques that use methods from linear algebra and number theory, starting at a rudimentary model to more complex algorithms that provide more security. The simpler model, Square Matrix Encryption, relies on the inverse of a square matrix, while a more complex algorithm, Rectangular Matrix Encryption, uses the pseudo-inverse of a rectangular matrix. This paper describes the nuances of both schemes and discusses what makes the latter stronger than the former. This research suggests using a special kind of generalized inverses in encryption that can turn into developing an even stronger encryption scheme. The implementation of all algorithms, which include various linear transformations and modular arithmetic on rational numbers, is done in Java.