Sparse polynomial interpolation: faster strategies over finite fields

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2024-04-27 DOI:10.1007/s00200-024-00655-5

Joris van der Hoeven, Grégoire Lecerf

引用次数: 0

Abstract

Consider a multivariate polynomial \(f \in K [x_1, \ldots , x_n]\) over a field K, which is given through a black box capable of evaluating f at points in \(K^n\), or possibly at points in \(A^n\) for any K-algebra A. The problem of sparse interpolation is to express f in its usual form with respect to the monomial basis. We analyze the complexity of various old and new algorithms for this task in terms of bounds D and T for the total degree of f and its number of terms. We mainly focus on the case when K is a finite field and explore possible speed-ups.

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稀疏多项式插值：有限域上的更快策略

考虑一个域 K 上的多元多项式（f \in K [x_1, \ldots , x_n]\），它通过一个黑盒子给出，黑盒子能够在 \(K^n\) 中的点对 f 进行求值，也可能在任意 K 代数 A 的 \(A^n\) 中的点对 f 进行求值。我们用 f 的总阶数和项数的边界 D 和 T 来分析这项任务的各种新旧算法的复杂性。我们主要关注 K 是有限域时的情况，并探索可能的提速方法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.