A Signature-Based Gröbner Basis Algorithm with Tail-Reduced Reductors (M5GB)

Gröbner bases are an important tool in computational algebra and, especially in cryptography, often serve as a boilerplate for solving systems of polynomial equations. Research regarding (efficient) algorithms for computing Gröbner bases spans a large body of dedicated work that stretches over the last six decades. The pioneering work of Bruno Buchberger in 1965 can be considered as the blueprint for all subsequent Gröbner basis algorithms to date. Among the most efficient algorithms in this line of work are signature-based Gröbner basis algorithms, with the first of its kind published in the late 1990s by Jean-Charles Faugère under the name $\texttt{F5}$. In addition to signature-based approaches, Rusydi Makarim and Marc Stevens investigated a different direction to efficiently compute Gröbner bases, which they published in 2017 with their algorithm $\texttt{M4GB}$. The ideas behind $\texttt{M4GB}$ and signature-based approaches are conceptually orthogonal to each other because each approach addresses a different source of inefficiency in Buchberger\u27s initial algorithm by different means. We amalgamate those orthogonal ideas and devise a new Gröbner basis algorithm, called $\texttt{M5GB}$, that combines the concepts of both worlds. In that capacity, $\texttt{M5GB}$ merges strong signature-criteria to eliminate redundant S-pairs with concepts for fast polynomial reductions borrowed from $\texttt{M4GB}$. We provide proofs of termination and correctness and a proof-of-concept implementation in C++ by means of the Mathic library. The comparison with a state-of-the-art signature-based Gröbner basis algorithm (implemented via the same library) validates our expectations of an overall faster runtime for quadratic overdefined polynomial systems that have been used in comparisons before in the literature and are also part of cryptanalytic challenges

Cyber Security Politics

This book examines new and challenging political aspects of cyber security and presents it as an issue defined by socio-technological uncertainty and political fragmentation. Structured along two broad themes and providing empirical examples for how socio-technical changes and political responses interact, the first part of the book looks at the current use of cyber space in conflictual settings, while the second focuses on political responses by state and non-state actors in an environment defined by uncertainties. Within this, it highlights four key debates that encapsulate the complexities and paradoxes of cyber security politics from a Western perspective – how much political influence states can achieve via cyber operations and what context factors condition the (limited) strategic utility of such operations; the role of emerging digital technologies and how the dynamics of the tech innovation process reinforce the fragmentation of the governance space; how states attempt to uphold stability in cyberspace and, more generally, in their strategic relations; and how the shared responsibility of state, economy, and society for cyber security continues to be re-negotiated in an increasingly trans-sectoral and transnational governance space. This book will be of much interest to students of cyber security, global governance, technology studies, and international relations

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described