A Bird’s-Eye View of Modern Symmetric Cryptography from Combinatorial Designs

In the past few decades, combinatorial design theory has grown to encompass a wider variety of investigations, many of which are not apparently motivated by any practical application. Rather, they are motivated by a desire to obtain a coherent and powerful theory of existence and properties of designs. Nevertheless, it comes as no surprise that applications in coding theory and communications continue to arise, and also that designs have found applications in new areas. Cryptography in particular has provided a new source of applications of designs, and simultaneously a field of new and challenging problems in design theory. In this paper, we present a number of applications of combinatorial designs in which the connection with modern symmetric (private-key) cryptography appears to be substantial and meaningful. We survey recent powerful private-key cryptosystems from special classes of combinatorial designs, i.e., orthogonal and Plotkin arrays, Hadamard matrices which are constructed from one and two circulant cores, which possess beautiful combinatorial properties. In addition, we present a new symmetric cryptosystem based on the famous Williamson construction for Hadamard matrices. Practical aspects of the cryptosystems, in terms of security and cryptanalysis, are analyzed and examples of real-time encryption and decryption are provided using cryptographic algorithms. We conclude by providing a state-of-the-art comparison of private-key block ciphers in the field of modern cryptography.