2021 EIMS Intensive Lecture Series

Abstracts

DAY 1. On Bent Functions

Boolean functions are important objects in discrete mathematics. They play a role in mathemat- ics and many domains of computer science. We will be mainly interested in their relationships with error-correcting codes and private-key cryptography.

The talk is devoted to special families of Boolean functions, viewed as significant objects in combinatorics and the information theory framework (cryptography and coding theory): the so-called bent functions.

Bent functions are maximally nonlinear Boolean functions. They are fantastic creatures in- troduced by O. Rothaus in the 1960s and initially studied by J. Dillon since 1974. For their own sake as interesting combinatorial objects, but also for their relations to coding theory (e.g., Reed-Muller codes, Kerdock codes, etc.), combinatorics (e.g., difference sets), design theory, sequence theory, and applications in cryptography (design of stream ciphers and of S-boxes for block ciphers), they have attracted a lot of research for four decades.

We present a profound presentation of bent functions and provide insight into them, including their characterizations, properties, constructions (primary and secondary constructions), and possible extensions. Besides, if time permits, we will also present an overview of one of their essential subclasses, namely, hyper-bent functions introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems revived interest in hyper-bent functions from the cryptographic point of view in 2016 (following the FSE conference-2016).

DAY 2. On Plateaued Functions

Boolean plateaued Boolean functions are very important cryptographic functions due to their various desirable cryptographic characteristics. We point out that plateaued functions are more general than bent functions (that is, functions with maximum nonlinearity).

Boolean plateaued, and p-ary plateaued functions have attracted a lot of attention in the litera- ture, and many activities on those functions have been carried out. The first aim of this lecture is to introduce and discuss Boolean plateaued and p-ary plateaued functions by presenting several various tools to handle the plateaued-ness property. The second aim of the talk is to increase our knowledge of plateaued functions in a more general context. We shall introduce and discuss generalized plateaued functions.

DAY 3. From (Optimal) Cryptographic Functions to (Optimal) Linear Codes over Finite Fields

In the modern world, the transmission and storage of information reliably and securely are of fundamental importance. There is an increasing amount of information exchange using com- munications over the Internet and mobile communication channels. Therefore, the security of communications and methods of information transfer will be essential for the exploitation of mobile communication and the conveyance of sensitive data. For many of these applications, security protocols are based on some cryptographic functions. Among these functions, we find optimal discrete functions with optimal cryptographic property from a cryptanalysis point of view. In this lecture, we will discuss how to use cryptographic functions to construct linear codes for various applications such as secret sharing schemes and secure two-party computation. After a brief introduction of linear codes and a profound description of the general constructions of linear codes from functions, we shall focus on the use of bent and plateaued functions and their variations to design optimal (or of good parameters) linear codes over finite fields.