## Research projects on cryptography

- Sedat Akleylek

The projects are suitable for all degrees (BSc, MSc, PhD)

**MDS Matrices over Binary Field Extensions and Their Implementations**

Block ciphers have diffusion and confusion layers. Maximum distance separable (MDS) matrices are the core component of the diffusion layers. MDS matrices have the maximum branch number. In this project, the aim is to summarize the search-based methods and direct construction methods to generate MDS matrices over binary field extensions. Another task is to compare the involutory MDS matrices in terms of XOR count after applying optimization methods.

**Distributed Storage Blockchain (DSB)**

The concept of distributed storage blockchain has been recently studied to reduce the storage cost of traditional blockchain systems. Network coding was adapted to the notion of distributed storage to reduce the storage space for distributed ledger in blockchain systems. One idea is to use Shamir's sharing scheme to decrease the storage of transactions. The task is to summarize the methods of secret sharing algorithms and linear codes for distributed storage with their application areas.

**Quantum Secure Digital Signature Schemes**

The task is to review the post-quantum digital signature schemes submitted to short signature and fast verification call. The focus will be on multivariate polynomial or lattice-based or code-based signature schemes.

## Research projects on coding and communications

- Irina Bocharova

**Signal sets indexing for future communications**

The development of energy-efficient signal sets for future communication standards requires optimization of signal indexing. Consider a set of signals (points in the Euclidean space) of a small dimension (at most 8). To each point, we assign an index represented by a binary sequence. To each pair of signals corresponds the Hamming distance between indices and the Euclidean distance between points. Good indexing avoids pairs that have a small Euclidean distance but a large Hamming distance. Keeping this in mind, a target function is derived. We need an efficient (maybe, AI) algorithm (program) to minimize the target function over the set of permutations of indices.

- Boris Kudryashov

** Covering sets for communications and post-quantum cryptology**

Some communication systems as well as post-quantum cryptology use error-correction algorithms based on so-called “coverings”. The covering problem is one of the fundamental problems in combinatorics. For a given length n and weight k of covering sequences we need to find the smallest possible set S of covering sequences, such that any w positions are covered (belong to) one of the sequences from S. In practice, we need a fast procedure for constructing S and it is desirable that all sequences in S can be obtained from its small subset by using simple operations such as permutations.

** Low-complexity quantum error-correcting coding **
The goal is to extend the expertise of the coding theory group in classic coding theory
to quantum codes. Some proprietary techniques developed in our projects are good candidates
for use in quantum data transmission. We invite students for participating in this research.

- Vitaly Skachek