Inverse matrices with applications in public-key cryptography

Abstract

The applications of non-square binary matrices span many domains including mathematics, error-correction coding, machine learning, data storage, navigation signals, and cryptography. In particular, they are employed in the McEliece and Niederreiter public-key cryptosystems. For the parity check matrix of these cryptosystems, a systematic non-square binary matrix (Formula presented.) with dimensions (Formula presented.), (Formula presented.), (Formula presented.), there exist (Formula presented.) distinct inverse matrices. This article presents an algorithm to generate these matrices as well as a method to construct a random inverse matrix. Then it is extended to non-square matrices in arbitrary fields. This overcomes the limitations of the Moore-Penrose and Gauss-Jordan methods. The application to public-key cryptography is also discussed.