Related papers: Multiplication in Cyclotomic Rings and its Applica…
Various post-quantum cryptography algorithms have been recently proposed. Supersingluar isogeny Diffie-Hellman key exchange (SIKE) is one of the most promising candidates due to its small key size. However, the SIKE scheme requires numerous…
We propose a numerical method based on the master field for large-$N$ reduced matrix models. While the master field is originally an infinite-dimensional matrix, in this method it is regularized to a finite dimension, with the requirement…
The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ…
Matrix multiplication is integral to various scientific and engineering disciplines, including machine learning, image processing, and gaming. With the increasing data volumes in areas like machine learning, the demand for efficient…
The lattice formulation provides a way to regularize, define and compute the Path Integral in a Quantum Field Theory. In this paper we review the theoretical foundations and the most basic algorithms required to implement a typical lattice…
We develop an automated framework for proving lower bounds on the bilinear complexity of matrix multiplication over finite fields. Our approach systematically combines orbit classification of the restricted first matrix and dynamic…
We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.
We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate polynomial…
We survey recent progress in computing with finitely generated linear groups over infinite fields, describing the mathematical background of a methodology applied to design practical algorithms for these groups. Implementations of the…
This work shows that minimizing the depth of a quantum circuit composed of commuting operations reduces to a vertex coloring problem on an appropriately constructed graph, where gates correspond to vertices and edges encode…
We give asymptotic expressions for the number of commuting matrices over finite fields. For this, we use product expansions for the corresponding generating functions.
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…
Many important applications across science, data analytics, and AI workloads depend on distributed matrix multiplication. Prior work has developed a large array of algorithms suitable for different problem sizes and partitionings including…
Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In…
The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of…
We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
In a previous paper we have introduced a class of multiplications of distributions in one dimension. Here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the…
In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation…
Unit group computations are a cryptographic primitive for which one has a fast quantum algorithm, but the required number of qubits is $\tilde O(m^5)$. In this work we propose a modification of the algorithm for which the number of qubits…