Discrete Functions and Their Applications in Cryptography and Mathematics

This project was concerned with the theory of functions over finite fields and its applications to cryptography and to various branches of mathematics. In cryptology the security of a cryptosystem is often based on hardness of solving certain mathematical problems. On the other hand, it turns out that to construct secure building blocks for cryptosystems it is often necessary to face mathematical challenges. The main aims of this project was to combine methods of algebra, combinatorics, coding theory, finite geometry, number theory, sequence design for construction of secure building blocks (that is, functions with special properties) for symmetric cryptography from one side, and to find more applications of discrete functions in the above mentioned areas of mathematics on the other side. During the project period the members of the project, Lilya Budaghyan and Nian Li, have achieved the following: According to the aims of this project the project members obtained results on construction and analysis of APN, AB, bent and negobent functions and their equivalence relations, construction of permutation polynomials over finite fields, construction of sequences with low correlation, construction of linear codes over finite fields or Galois rings with good parameters. The PI, Lilya Budaghyan, has organized 6 international workshops and conferences: BFA 2014 and BFA 2017, BFA 2018 (Boolean Functions and Their Applications) and MMC 2017 (Mathematical Methods for Cryptography), WAIFI 2018 (Finite Fields and Their Applications) and Emil Artin Conference. The project members have published 2 monograph, 24 papers and 5 books as a co-editor. In 2016 the PI obtained a 23 million NOK project grant "Optimal Boolean functions" from Bergen Research Foundation to continue and deepen the research started in the current NFR project. In 2018, after finishing the work in this project, the second researcher of this project, Nian Li, started as an associate professor at Hubei University in China. The PI has been a member of program committees of international conferences: 1. Int. Conference on Sequences and Their Applications SETA 2018, Hong Kong. 3. International Students' Olympiad in Cryptography NSUCRYPTO. 4. Int. Workshop on the Arithmetic of Finite Fields, WAIFI 2018, Bergen, Norway, (program co-chair). 5. Int. Workshop on Coding and Cryptography WCC 2017, Saint Petersburg, Russia. 6. Second Int. Conference on Codes, Cryptology and Information Security, C2SI 2017, Rabat, Morocco. 7. The 9th Norwegian Information Security Conference, NISK 2016, Bergen, Norway. The PI has given invited talks: 1. Workshop Promoting cooperation between researchers in Armenia and Scientific Diaspora in ICT and related research fields, within CSIT 2017 international conference, Yerevan, Armenia, 2017. 2. Moscow State University, Moscow, Russia, May 2017. 3. Armenian Mathematical Union, Yerevan, Armenia, May 2017. 4-5. Information Security and Protection of Information Technologies workshop, ISPIT-2017 and ISPIT-2016, St Petersburg, Russia, Apr. 2016 and Apr. 2017. 6. Norwegian - Slovakian Workshop, Bergen, Norway, Feb. 2016. 7. Novosibirsk State University, Russia, Oct. 2018. 8. Steklov Institute of Mathematics, Novosibirsk, Russia, Oct. 2018. 9. Emil Artin Conference, Yerevan, Armenia, May 2018. The PI served as an co-editor of special issues of international journals: 1. "Cryptography and Communications" (Springer) 2017/2018 for the Special Issue on Boolean Functions and Their Applications. 2. "Cryptography and Communications" 2017/2018 for the Special Issue on Mathematical Methods for Cryptography. 3. "Cryptography and Communications" 2014/2015 for the Special Issue on Boolean Functions and Their Applications (130 pages). 4. "Designs, Codes and Cryptography" (Springer) 2013/2014 for the Special Issue on Coding and Cryptography (332 pages). 5. "Lecture Notes in Computer Science" 2018/2019 for Post-proceedings of the International Workshop on Finite Fields and Their Applications. The PI has been a main supervisor of PhD students Bo Sun (2016-2018), Irene Villa (since 2016), Nikolay Kaleyski (since 2017), Diana Davidova (since 2017), and a co-supervisor of Dan Zhang (since 2017). The PI has participated as collaborator in the projects: 1. NRC project "Modern Methods and Tools for Symmetric-Key Cryptology" (24 MNOK) for period 2015 - 2020; 2. "Vulnerabilities of systems for protection against SCA attacks" at Saint-Petersburg State University of Information Technologies, Mechanics and Optics, Saint-Petersburg, Russia, for period from April to July 2016; 3. "Cryptography brings security and freedom" (EEA Grant for Slovakian-Norwegian cooperation), period 2015 - 2016.

Many results on discrete functions have been obtained which have been published in 2 monograph and 24 papers. In addition 5 books as a co-editor have been published, 6 workshops organized, membership in 7 program committees of international conferences, 9 invited talks at international conferences and foreign universities, 5 PhD students have been trained (one of which with graduation during the project period), a 23 million NOK BFS project obtained, participation in 3 projects as a collaborator.

The aim of this project is to address essential problems related to discrete functions. Solutions of these problems may improve the reliability and security of modern communication systems and could make a big impact to many branches of mathematics. That is, the research involved in this project would have both theoretical and practical significance. Discrete functions are mappings from one finite set to another one and an important particular case of such functions is the case of Boolean functions, whic h are ubiquitous, occurring at the heart of virtually all known digital systems: computers, telecommunications and cryptographic primitives, for example, all depend on the theory of Boolean functions. Moreover, functions with optimal cryptographic propert ies define optimal in certain sense objects in coding theory, combinatorics, algebra, finite geometry, sequence design, quantum information theory et al. Hence, construction and analysis of optimal cryptographic functions is connected with important mathe matical problems and solution of these problems would make a valuable contribution to both mathematics and information theory. On the other hand, it's certain that the potential of the theory of discrete functions is not used to its full extent and we als o aim to discover its further applications.