Abstract

The purpose of this talk is to describe some algebraic results and applications for Multivariate Cryptography. In our work we chose a general approach for multivariate quadratic systems over finite fields and we analyzed some algebraic invariants of Oil &Vinegar and mixed quadratic systems both in the under-determined and over-determined cases. This allowed us to estimate both the first fall degree and the degree of regularity, that strongly impact the security of the system. In addition, we introduce a specific definition of a degree of regularity for an OV homogeneous system, and, we show that, when working in finite fields, we may have saturation of smaller spaces in degrees lower than such degree. The first degree for which this saturation happens corresponds the first fall degree. We gave explicit formulas for the dimensions of such spaces, both for purely OV systems and mixed quadratic ones and we got some bounds for the first fall degree. We did an application of mixed quadratic systems for build a system for key-exchange with trapdoor, called OliVier. Finally, there will be an analysis on symmetric polynomials for cryptographic applications. In particular, we focus on two quadratic maps that could be used for digital signatures and, in both case, we obtained that they are equivalent to two schemes already known, that are the Stepwise Triangular Systems (STS) and the UOV ones.

Bio

Rosa Fera is a third-year PhD student in Algebra and Cryptography at the Department of Information and Electrical Engineering of University of Cassino and Southern Lazio (Italy). She is graduated in Mathematics. Her interest in Cryptography started when she wrote her Master's thesis in which she studied Group Theory applied to Cryptography. After the Master's degree, she spent 6 month for a stage at ST Microelectronics (Marcianise, Italy) where she worked on Fully Homomorphic Encryption and how to apply it on secure elements. In March 2023, she started her PhD at University of Cassino and Southern Lazio. The main research topic of her PhD is Multivariate Cryptography.