关于有限域中 r 次根的计算

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2024-08-05 DOI:10.1016/j.ffa.2024.102479

Gook Hwa Cho , Soonhak Kwon

引用次数: 0

摘要

设 q 是一个质数的幂，使得 q≡1(modr)。设 c 是 Fq 上的 r 次幂残差。在本文中，我们提出了一个新的 r-th 根公式，它概括了 G.H. Cho 等人的立方根算法，并对 Williams 基于 Cipolla-Lehmer 的程序进行了改进。我们的算法基于不可还原多项式 h(x)=xr+(-1)r+1(b+(-1)rr)(x-1) 所产生的递推关系，其中 b=c+(-1)r+1r 对于 r>1 只需要 O(r2logq+r4) 次乘法。

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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On the computation of r-th roots in finite fields

Let q be a power of a prime such that $q \equiv 1 (\mod r)$ . Let c be an r-th power residue over $F_{q}$ . In this paper, we present a new r-th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial $h (x) = x^{r} + {(- 1)}^{r + 1} (b + {(- 1)}^{r} r) (x - 1)$ with $b = c + {(- 1)}^{r + 1} r$ requires only $O (r^{2} \log q + r^{4})$ multiplications for $r > 1$ . The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.