Innovative construction of involutory MDS matrices via self-reciprocal generator polynomials derived from reed-solomon codes

Abstract

Recursive MDS matrices over finite fields optimize diffusion and enable efficient implementation in block ciphers. A key challenge is designing involutory MDS matrices to unify encryption and decryption, reducing costs. Recursive MDS matrices can meet these requirements. In this paper, we present a direct construction method for self-reciprocal recursive MDS matrices of arbitrary sizes, derived from self-reciprocal generator polynomials of Reed-Solomon codes, to generate corresponding involutory MDS matrices. The method for deriving self-reciprocal recursive MDS matrices from Reed-Solomon codes is straightforward. Additionally, we identify self-reciprocal generator polynomials of RS codes over the general finite field , where  and  is a prime number, including cases where  is an odd prime. Involutory MDS matrices derived from self-reciprocal matrices are highly efficient for hardware and software, making them ideal for modern cryptographic applications.