A Bird’s-Eye View of Optimal Codes and Symmetric Cryptography from Combinatorial Designs
Dimitris E. Simos, Department of Mathematics, National Technical University of Athens,
Abstract: 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 in computer science, and simultaneously a field of new and challenging problems in design theory.
In this lecture, we present a number of applications of combinatorial designs in which the connection with classes of optimal codes and modern symmetric (private-key) cryptography appears to be substantial and meaningful. In the first part, we present some new results for self-dual codes and quasi-cyclic codes and exemplify some of their advantages in terms of encoding and decoding. In the continuum, we survey recent powerful private-key cryptosystems from special classes of combinatorial designs, that posses beautiful combinatorial properties. 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.