当前位置: X-MOL 学术Finite Fields Their Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Analysis of generalized continued fraction algorithms over polynomials
Finite Fields and Their Applications ( IF 1 ) Pub Date : 2021-04-23 , DOI: 10.1016/j.ffa.2021.101849
Valérie Berthé , Hitoshi Nakada , Rie Natsui , Brigitte Vallée

We study and compare natural generalizations of Euclid's algorithm for polynomials with coefficients in a finite field. This leads to gcd algorithms together with their associated continued fraction maps. The gcd algorithms act on triples of polynomials and rely on two-dimensional versions of the Brun, Jacobi–Perron and fully subtractive continued fraction maps, respectively. We first provide a unified framework for these algorithms and their associated continued fraction maps. We then analyse various costs for the gcd algorithms, including the number of iterations and two versions of the bit-complexity, corresponding to two representations of polynomials (the usual and the sparse one). We also study the associated two-dimensional continued fraction maps and prove the invariance and the ergodicity of the Haar measure. We deduce corresponding estimates for the costs of truncated trajectories under the action of these continued fraction maps, obtained thanks to their transfer operators, and we compare the two models (gcd algorithms and their associated continued fraction maps). Proving that the generating functions appear as dominant eigenvalues of the transfer operator allows indeed a fine comparison between the models.



中文翻译:

多项式的广义连续分数算法分析

我们研究并比较了在有限域中具有系数的多项式的Euclid算法的自然概括。这导致gcd算法及其关联的连续分数图。gcd算法作用于多项式的三元组,分别依赖于Brun,Jacobi–Perron和完全减法连续分数图的二维版本。我们首先为这些算法及其相关的连续分数图提供一个统一的框架。然后,我们分析gcd算法的各种成本,包括迭代次数和两种形式的位复杂度,分别对应多项式的两种表示形式(通常的和稀疏的一种)。我们还研究了相关的二维连续分数图,并证明了Haar测度的不变性和遍历性。我们推导了在这些连续分数图的作用下截断轨迹的成本的相应估计值,这要归功于它们的转移算子,并且我们比较了两种模型(gcd算法及其关联的连续分数图)。证明生成函数显示为转移算子的主要特征值,确实可以在模型之间进行很好的比较。

更新日期:2021-04-23
down
wechat
bug