**Nikolay Kaleyski**

**Ph.D. started in: **2017**Expected year of graduation: **2021**COINS consortium member: **University of Bergen**Supervised by:** Lilya Budaghyan**Links:** **Research area: **Cryptography**Project title:** Properties of optimal Boolean functions**Project description:** Boolean functions constitute an integral part of various encryption algorithms (such as block ciphers, in the context of which they are typically referred to as “S-boxes” or “substitution boxes” since they act as components substituting one sequence of bits for another) and as such play a central role in the design and analysis of such algorithms.

The security of the encryption depends directly on certain properties of the underlying Boolean functions, such as their non-linearity and differential uniformity whose values determine the resistance against linear and differential attacks, respectively. Different classes of Boolean functions, e.g. almost perfect non-linear (APN) functions, almost bent (AB) functions, bent functions, planar functions, etc. are defined so that the functions contained therein are optimal with respect to some given desirable quality, e.g. non-linearity. However, many questions of practical importance arise which are not answered by the definitions per se, most notably the question of how to actually construct functions of a given type (APN or AB, for example); furthermore, while many constructions are known e.g. in the case of APN functions, new ones continue to be discovered and it is thus not known whether other APN constructions exist and how they might be performed; in addition, many open questions remain pertaining to the properties of the constructions that have already been found as well. The goal of the project, therefore, is to investigate these various classes (APN, AB, bent, etc) of optimal Boolean functions with the purpose of better understanding their particular properties as well as finding new methods of constructing such functions.

**Publications:**

- Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**(2021). Towards a deeper understanding of APN functions and related longstanding problems - Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**, Constanza Susana Riera, Pantelimon Stanica (2021). On the behavior of some APN permutations under swapping points - Diana Davidova,
**Nikolay Stoyanov Kaleyski**(2021). Generalization of a class of APN binomials to Gold-like functions **Nikolay Stoyanov Kaleyski**(2021). Deciding EA-equivalence via invariants**Nikolay Stoyanov Kaleyski**(2021). Invariants for EA- and CCZ-equivalence of APN and AB functions- Lilya Budaghyan, Claude Michael Carlet, Tor Helleseth,
**Nikolay Stoyanov Kaleyski**(2020). On the Distance Between APN Functions - Lilya Budaghyan, Tor Helleseth,
**Nikolay Stoyanov Kaleyski**(2020). A New Family of APN Quadrinomials - Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**, Constanza Susana Riera, Pantelimon Stanica (2020). On the sensitivity of some permutation APN functions to swapping points - Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**, Constanza Susana Riera, Pantelimon Stănică (2020). Partially APN functions with APN-like polynomial representations - Yuyin Yu,
**Nikolay Stoyanov Kaleyski**, Lilya Budaghyan, Yongqiang Li (2020). Classification of quadratic APN functions with coefficients in F2 for dimensions up to 9 - Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**, Soonhak Kwon, Constanza Susana Riera, Pantelimon Stănică (2019). Partially APN Boolean functions and classes of functions that are not APN infinitely often **Nikolay Stoyanov Kaleyski**(2019). Changing APN functions at two points- Lilya Budaghyan,
**Nikolay Stoyanov Kaleyski**, Soonhak Kwon, Constanza Susana Riera, Pantelimon Stanica (2018). Partially APN Boolean functions

**Events attended with COINS funding:**

- BFA 2019, Florence, Italy, 2019
- COINS Finse winter school, Finse, Norway, 2018
- COINS Ph.D student seminar, Longyearbyen, Svalbard, Norway, 2018
- Emil Artin International Conference, Yerevan, the Republic of Armenia, 2018
- BFA 2018 workshop, Loen, Norway, 2018
- NISK 2018, Longyearbyen, Svalbard, Norway, 2018
- Mathematical Methods for Cryptography , Svolvær, Norway, 2017