Cryptographic functions defined over finite fields play a central role in the security of symmetric crypto-systems. They are one of the cornerstones of our cryptographic landscape today and are used to ensure security for a significant fraction of our daily communication. Notably, for practice use, they are used as the pseudo-random generators of stream ciphers and considered crucial functions; they are also widely applied to block ciphers' design in cryptography (where they are used to create the so-called substitution boxes, or S-boxes, whose input and output are then both sequences of bits). We shall mainly focus on some crucial cryptographic parameters for (Boolean) functions, such as nonlinearity. We shall present some achievements on cryptographic functions where the aspects of high nonlinearity will take precedence over others and discuss open problems on this topic and directly related ones. We will introduce related functions, provide some insight into them, and discuss some problems to increase our knowledge (particularly in design) about their corpus. We will also be interested in exploring more cryptographic functions over finite fields to increase our knowledge, present mathematical tools to handle them and investigate designing them.
Cryptographic functions: a selection of some achievements and exploring of new research advances and directions*
Sihem Mesnager, Universities of Paris VIII and Paris XIII-LAGA, and Telecom Paris, Polytechnic Institute
Authors: S. Mesnager
2022 AWM Research Symposium
Mathematical Aspects of Cryptography