Related papers: Middle-Solving Grobner bases algorithm for cryptan…
Numerous conceptually important quantum algorithms rely on a black-box device known as an oracle, which is typically difficult to construct without knowing the answer to the problem that the algorithm is intended to solve. A notable example…
Non-linear polynomial systems over finite fields are used to model functional behavior of cryptosystems, with applications in system security, computer cryptography, and post-quantum cryptography. Solving polynomial systems is also one of…
We demonstrate that the most popular variants of all common algebraic multidimensional rootfinding algorithms are unstable by analyzing the conditioning of subproblems that are constructed at intermediate steps. In particular, we give…
The graph continual release model of differential privacy seeks to produce differentially private solutions to graph problems under a stream of edge updates where new private solutions are released after each update. Thus far, previously…
We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the…
We give an example where the number of elements of a Groebner basis in a Boolean ring is not polynomially bounded in terms of the bitsize and degrees of the input.
In this paper, we make a contribution to the computation of Gr\"obner bases. For polynomial reduction, instead of choosing the leading monomial of a polynomial as the monomial with respect to which the reduction process is carried out, we…
In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of…
The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields. This, however, is only attainable in those cases where analytical solutions exist. Many others invoke the use of numerical…
We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem.…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
We give upper bounds for the solving degree and the last fall degree of the polynomial system associated to the HFE (Hidden Field Equations) cryptosystem. Our bounds improve the known bounds for this type of systems. We also present new…
We present Groebner.jl, a Julia package for computing Groebner bases with the F4 algorithm. Groebner.jl is an efficient, portable, and open-source software. Groebner.jl works over integers modulo a prime and over the rationals, supports…
Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…
In this letter, we delve into a scenario where a user aims to compute polynomial functions using their own data as well as data obtained from distributed sources. To accomplish this, the user enlists the assistance of $N$ distributed…
In this paper we present an improved version of HF-hash, viz., GB-hash : Hash Functions Using Groebner Basis. In case of HF-hash, the compression function consists of 32 polynomials with 64 variables which were taken from the first 32…
Let K be a field with a valuation and let S be the polynomial ring S:= K[x_1,..., x_n]. We discuss the extension of Groebner theory to ideals in S, taking the valuations of coefficients into account, and describe the Buchberger algorithm in…
Here we suggest a modification of Grover's algorithm, based on a multiphase oracle which marks each solution with a different phase when there is more than one solution. Such a modification can be used to maintain a high probability of…
Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e., solving minimal…