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自从1985年椭圆曲线密码被提出后,其理论和应用研究都受到了广泛关注.椭圆曲线密码体制的安全性基于椭圆曲线离散对数问题的困难性.由于计算一般椭圆曲线中离散对数的算法都是指数时间的,椭圆曲线密码体制能够以更小的密钥尺寸来满足与其他公钥密码体制相同的安全性要求,从而特别适用于计算和存储能力受限的领域,许多标准化组织也相继提出了椭圆曲线上的公钥加密、密钥协商、数字签名协议的标准.利用Schoof’s算法或复乘方法,人们可以很容易构造出密码学所需的椭圆曲线.通常推荐使用的椭圆曲线都定义在特征为2的有限域或素域上.为了加速有限域的运算,部分学者提议使用非素域有限域.然而对于非素域有限域上椭圆曲线中离散对数,基于求和多项式的指标计算法和Weil下降方法有可能比Pollard’s Rho等一般性算法快.因此研究这些算法对椭圆曲线离散对数问题困难性的削弱程度以及相应的弱曲线特点对椭圆曲线密码学的安全应用有重大意义.本文将对解椭圆曲线离散对数问题的方法和研究进展做简单综述.
Since the introduction of elliptic curve cryptography in 1985, both theoretical and applied research have drawn much attention. The security of elliptic curve cryptosystem is based on the difficulty of discrete logarithm problem of elliptic curve. Since the algorithm of computing discrete logarithm in elliptic curve Are both exponential, elliptic curve cryptosystems can meet the same security requirements as other public-key cryptosystems with smaller key sizes, making them particularly well-suited for use in areas where computing and storage capabilities are limited, and many standardization organizations have also followed Proposed the standard of the public key encryption, the key agreement and the digital signature protocol on the elliptic curve.Using Schoof’s algorithm or complex multiplication method, one can easily construct the elliptic curve required by cryptography.Elliptic curves usually recommended are defined In the finite field or prime domain with characteristic 2, some scholars propose to use the non-prime domain finite field in order to speed up the operation of the finite field. However, for the discrete logarithm of the elliptic curve on the non-prime domain, the index based on the summation polynomial Computational methods and Weil descent methods are likely to be faster than the general ones such as Pollard’s Rho, etc. Therefore, Powder on the degree of difficulty of weakening and the corresponding number of weak curve characteristic problem of great importance to the security applications of elliptic curve cryptography. This article will de-elliptic curve discrete logarithm problem methods and research progress on to do a simple review.