Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

New Techniques for Polynomial System Solving

Since any encryption map may be viewed as a polynomial map between finite dimensional vector spaces over finite fields, the security of a cryptosystem can be examined by studying the difficulty of solving large systems of multivariate polynomial equations. Therefore, algebraic attacks lead to the task of solving polynomial systems over finite fields. In this thesis, we study several new algebraic techniques for polynomial system solving over finite fields, especially over the finite field with two elements. Instead of using traditional Gröbner basis techniques we focus on highly developed methods from several other areas like linear algebra, discrete optimization, numerical analysis and number theory. We study some techniques from combinatorial optimization to transform a polynomial system solving problem into a (sparse) linear algebra problem. We highlight two new kinds of hybrid techniques. The first kind combines the concept of transforming combinatorial infeasibility proofs to large systems of linear equations and the concept of mutants (finding special lower degree polynomials). The second kind uses the concept of mutants to optimize the Border Basis Algorithm. We study recent suggestions of transferring a system of polynomial equations over the finite field with two elements into a system of polynomial equalities and inequalities over the set of integers (respectively over the set of reals). In particular, we develop several techniques and strategies for converting the polynomial system of equations over the field with two elements to a polynomial system of equalities and inequalities over the reals (respectively over the set of integers). This enables us to make use of several algorithms in the field of discrete optimization and number theory. Furthermore, this also enables us to investigate the use of numerical analysis techniques such as the homotopy continuation methods and Newton's method. In each case several conversion techniques have been developed, optimized and implemented. Finally, the efficiency of the developed techniques and strategies is examined using standard cryptographic examples such as CTC and HFE. Our experimental results show that most of the techniques developed are highly competitive to state-of-the-art algebraic techniques

Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing