Implementation of point-counting algorithms on genus 2 hyperelliptic curves based on the birthday paradox

IF 0.2 Q4 MATHEMATICS, APPLIED

Prikladnaya Diskretnaya Matematika Pub Date : 2022-01-01 DOI:10.17223/20710410/55/9

N. Kolesnikov

引用次数: 0

Abstract

Our main contribution is an efficient implementation of the Gaudry - Schost and Galbraith - Ruprai point-counting algorithms on Jacobians of hyperelliptic curves. Both of them are low memory variants of Matsuo - Chao - Tsujii (MCT) Baby-Step Giant-Step-like algorithm. We present an optimal memory restriction (a time-memory tradeoff) that minimizes the runtime of the algorithms. This tradeoff allows us to get closer in practical computations to theoretical bounds of expected runtime at 2.45√N and 2.38a√N for the Gaudry - Schost and Galbraith - Ruprai algorithms, respectively. Here N is the size of the 2-dimensional searching space, which is as large as the Jacobian group order, divided by small modulus m, precomputed by using other techniques. Our implementation profits from the multithreaded regime and we provide some performance statistics of operation on different size inputs. This is the first open-source parallel implementation of 2-dimensional Galbraith - Ruprai algorithm.

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基于生日悖论的2属超椭圆曲线点计数算法的实现

我们的主要贡献是在超椭圆曲线的雅可比矩阵上有效地实现了Gaudry - Schost和Galbraith - Ruprai点计数算法。这两种算法都是Matsuo - Chao - Tsujii (MCT) Baby-Step - Giant-Step-like算法的低内存变体。我们提出了一个最优内存限制(时间-内存权衡)，使算法的运行时间最小化。这种权衡使我们在实际计算中更接近Gaudry - Schost和Galbraith - Ruprai算法的预期运行时间分别为2.45√N和2.38a√N的理论界限。这里N是二维搜索空间的大小，等于雅可比群阶，除以用其他技术预先计算的小模m。我们的实现得益于多线程机制，我们提供了不同大小输入操作的性能统计数据。这是第一个二维Galbraith - Ruprai算法的开源并行实现。

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

Prikladnaya Diskretnaya Matematika MATHEMATICS, APPLIED-

CiteScore

0.60

自引率

50.00%

发文量

期刊介绍： The scientific journal Prikladnaya Diskretnaya Matematika has been issued since 2008. It was registered by Federal Control Service in the Sphere of Communications and Mass Media (Registration Witness PI № FS 77-33762 in October 16th, in 2008). Prikladnaya Diskretnaya Matematika has been selected for coverage in Clarivate Analytics products and services. It is indexed and abstracted in SCOPUS and WoS Core Collection (Emerging Sources Citation Index). The journal is a quarterly. All the papers to be published in it are obligatorily verified by one or two specialists. The publication in the journal is free of charge and may be in Russian or in English. The topics of the journal are the following: 1.theoretical foundations of applied discrete mathematics – algebraic structures, discrete functions, combinatorial analysis, number theory, mathematical logic, information theory, systems of equations over finite fields and rings; 2.mathematical methods in cryptography – synthesis of cryptosystems, methods for cryptanalysis, pseudorandom generators, appreciation of cryptosystem security, cryptographic protocols, mathematical methods in quantum cryptography; 3.mathematical methods in steganography – synthesis of steganosystems, methods for steganoanalysis, appreciation of steganosystem security; 4.mathematical foundations of computer security – mathematical models for computer system security, mathematical methods for the analysis of the computer system security, mathematical methods for the synthesis of protected computer systems;[...]