Research database

Cryptographic systems play an increasingly important role in our society in order to secure privacy and confidentiality of communications. A deep cryptanalysis of these systems is really crucial, with the aim of studying their potential weaknesses and highlighting the conditions that allow to attack them. In this way, we are able to provide decisive advices for designing, implementing and using effectively safe ciphers. The continuous advancement of the modern cryptographic systems makes cryptanalysis one of the most active research fields that deserves to be widely studied. It is well known that cryptography and cryptanalysis are mainly based on pure mathematics and specifically on algebra and number theory, think to the importance of elliptic curves, finite fields, group theory and polynomial systems. This project aims at obtaining new results in the field of cryptanalysis with innovative methodologies that may open new research directions. Attention will be paid to cryptanalysis of both symmetric and asymmetric schemes, with the goal to also apply the theoretical results to the lightweight cryptographic systems and post-quantum systems currently submitted to the NIST competitions. Lines of research are the following: - Algebraic cryptanalysis for stream, block and public key ciphers is essentially based on solving involved polynomial systems over finite fields. The complexity to perform such solving is an assessment of the security of the corresponding ciphers. Many approaches are possible to such tasks such as symbolic computation, satisfiability checking, binary decision diagrams, etc. In this project we will develop symbolic computation, namely Grobner bases, having in mind that smart guess-and-determine strategies are a key tool to provide not only theoretical but also practical complexity. - The conjugacy class of elementary abelian regular subgroups of the symmetric group has recently drawn the attention of symmetric cryptographers, as these subgroups and their normalizers may be used to detect weaknesses in symmetric-encryption methods. For this reason one of the goals of this project is to obtain a deeper knowledge of the structure of such subgroups suitable for cryptanalysis - In the elliptic curve cryptography, anomalous curves are not recommended due to attacks based on a lift in the field of p-adic numbers and a p-adic elliptic logarithm. These strategies may inspire similar attacks for other classes of curves, in particular studying if the lift of a curve into rings might imply a flaw in cryptographic protocols. Other classical asymmetric schemes can be attacked exploiting properties of continued fractions, we aim at creating new attacks using multidimensional continued fractions - We propose to search for new structured groups with algorithms for the Hidden Subgroup Problem that are time-efficient, space-efficient, or both, with particular attention to cases related to the security of classical and post-quantum cryptosystems