Sub-cubic Change of Ordering for Gröner Basis: A Probabilistic Approach

International audienceThe usual algorithm to solve polynomial systems using Gröbner bases consists of two steps: first computing the DRL Gröbner basis using the F5 algorithm then computing the LEX Gröbner basis using a change of ordering algorithm. When the Bézout bound is reached, the bottleneck of the total solving process is the change of ordering step. For 20 years, thanks to the FGLM algorithm the complexity of change of ordering is known to be cubic in the number of solutions of the system to solve. We show that, in the generic case or up to a generic linear change of variables, the multiplicative structure of the quotient ring can be computed with no arithmetic operation. Moreover, given this multiplicative structure we propose a change of ordering algorithm for Shape Position ideals whose complexity is polynomial in the number of solutions with exponent ω where 2 ≤ ω < 2.3727 is the exponent in the complexity of multiplying two dense matrices. As a consequence, we propose a new Las Vegas algorithm for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic (i.e. with exponent ω) complexity and whose total complexity is dominated by the complexity of the F5 algorithm. In practice we obtain significant speedups for various polynomial systems by a factor up to 1500 for specific cases and we are now able to tackle some instances that were intractable

Newton polytopes and numerical algebraic geometry

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry.Comment: 150 pages, 65 figures, contains content from arXiv:1811.12279 and arXiv:2001.0422

Newton Polytopes and Numerical Algebraic Geometry

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry

Grained integers and applications to cryptography

To meet the requirements of the modern communication society, cryptographic techniques are of central importance. In modern cryptography, we try to build cryptographic primitives, whose security can be reduced to solving a particular number theoretic problem for which no fast algorithmic method is known by now. Thus, any advance in the understanding of the nature of such problems indirectly gives insight in the analysis of some of the most practical cryptographic techniques. In this work we analyze exactly this aspect much more deeply: How can we use some of the purely theoretical results in number theory to answer very practical questions on the security of widely used cryptographic algorithms and how can we use such results in concrete implementations? While trying to answer these kinds of security-related questions, we always think two-fold: From a cryptographic, security-ensuring perspective and from a cryptanalytic one. After we outlined -- with a special focus on the historical development of these results -- the necessary analytic and algorithmic foundations of number theory, we first delve into the question how point addition on certain elliptic curves can be done efficiently. The resulting formulas have their application in the cryptanalysis of crypto systems that are insecure if factoring integers can be done efficiently. The rest of the thesis is devoted to the study of integers, all of whose prime factors are neither too small nor too large. We show with the help of two applications how one can use the properties of such kinds of integers to answer very practical questions in the design and the analysis of cryptographic primitives: The optimization of a hardware-realization of the cofactorization step of the General Number Field Sieve and the analysis of different standardized key-generation algorithms

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Radial projection statistics: a different angle on tilings

Jakobi T. Radial projection statistics: a different angle on tilings. Bielefeld: Universität Bielefeld; 2017

The Design and Implementation of a High-Performance Polynomial System Solver

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation. Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical property of the particular input system’s solution set. Despite these challenges, we have effectively applied parallel computing to triangular decomposition through the layering and cooperation of many parallel code regions. This parallel computing is supported by our generic object-oriented framework based on the dynamic multithreading paradigm. Meanwhile, the required polynomial algebra is sup- ported by an object-oriented framework for algebraic types which allows type safety and mathematical correctness to be determined at compile-time. Our software is implemented in C/C++ and have extensively tested the implementation for correctness and performance on over 3000 polynomial systems that have arisen in practice. The parallel framework has been re-used in the implementation of Hensel factorization as a parallel pipeline to compute roots of a polynomial with multivariate power series coefficients. Hensel factorization is one step toward computing the non-trivial limit points of quasi-components