密码学中的椭圆曲线理论研究

Elliptic curve cryptography (ECC) and elliptic curve factorization method (ECM) are the outstanding applications of elliptic curve theory in cryptography.Elliptic curve discrete logarithm problem (ECDLP) is the security foundation of ECC, which determines the existence and use of ECC; ECM is one of the fastest integer factorization algorithm. This project will study ECDLP and ECM, and the main research contents include analyzing whether the five elliptic curves defined over finite fields GF(2^m),m=176,208,272,304,368 can defense the generalized GHS attack;extending the new index calculus proposed by the best paper on Eurocrypt 2012 to more curves;constructing a new iteration function for Pollard rho algorithm; constructing new Edwards curves or curves with genus 2 over the extensions of rational number field to improve the efficient of ECM. This project have important theoretical and practical significance which will help us grasp the progress of the international research correctly and determine the results of international research accurately so as to enhance our capability of domestically design and innovation.

椭圆曲线密码（ECC）和椭圆曲线分解法（ECM）是椭圆曲线理论在密码学中杰出的应用。椭圆曲线离散对数问题（ECDLP）是ECC密码安全性的基石，它决定了ECC密码的存在和使用；ECM是分解大整数的最快算法之一。本项目研究ECDLP和ECM,主要研究内容有：分析ANSI X9.62标准中定义在有限域GF(2^m),m=176,208,272,304,368上的五条椭圆曲线是否能抵抗广义GHS攻击；把2012年欧洲密码年会最佳论文提出的新指标计算法（index calculus）推广应用到更多的曲线；构造新的迭代函数用于Pollard rho算法；在有理数扩域上构造Edwards曲线或亏格为２的曲线用于提高ECM效率。本项目对我们正确把握和准确判断国际相关研究的进展和结果，增强自主设计和创新能力，具有极其重要的理论和现实意义。

本项目围绕密码学中的椭圆曲线理论展开研究，涉及的研究内容包括：椭圆曲线分解法、椭圆曲线点乘以及双线性对。. 在自然科学基金的支持下，我们取得了一系列的成果，主要包括：构造了可以加快椭圆曲线分解法的Edwards曲线；分析了二元扩域和三元扩域椭圆曲线的Montgomery 阶梯算法；推广并优化了Montgomery 阶梯算法；加快了双线性对的计算效率；使用故障攻击分析了椭圆曲线密码和双线性对密码，并给出了防御措施。