19,757 research outputs found
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
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
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
Improved Agreeing-Gluing Algorithm
In this paper we study the asymptotical complexity of solving a system of sparse algebraic equations over finite fields. An equation is called sparse if it depends
on a bounded number of variables. Finding
efficiently solutions to the system of such equations is an underlying hard problem in
the cryptanalysis of modern ciphers. New deterministic
Improved Agreeing-Gluing Algorithm is introduced.
The expected
running time of the Algorithm on uniformly random instances of the problem is rigorously estimated. The estimate is at present the best theoretical
bound on the complexity of solving average instances of the
problem. In particular, this is a significant improvement over those in our earlier papers [20,21].
In sparse Boolean equations a gap between the present worst case and the average time complexity of the problem has significantly increased. Also we formulate Average Time Complexity Conjecture. If proved that will have far-reaching consequences in the field of cryptanalysis and in computing in general
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
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