Effective Methods for Hyperelliptic and Cubic Function Fields

研究了椭圆、超椭圆、三次曲线及其函数域理论中的各种有效方法。这些领域的经典问题是计算类数、调节器和离散对数，以及确定雅可比的基数。第一部分是关于超椭圆曲线和三次曲线的算术不变量。特别是，这位研究人员和他的同事们希望推进大型素数域上这些曲线的计点方法。有效的方法利用模方程、Zeta函数的零点分布、Hasse-Witt矩阵、欧拉积的近似、优化算法等。第二部分是关于椭圆曲线的Weil下降方法或其他Galois下降方法。Weil下降方法是将复合有限域上的椭圆曲线离散对数问题归结为真子域上的离散对数问题的一种方法。这导致了一种有效的方法来将有限域上的ECDLP的任何实例归结为子域上的超椭圆曲线的雅可比中的离散对数问题的实例。由于后一个问题的次指数时间算法是已知的，这表明该方法对于破解某些椭圆曲线密码方案是多么重要。类似的思想也适用于亏格大于一的曲线。所提出的研究属于数论和代数几何之间的界面。在理论方面，提出了代数函数场和曲线的理论。同时，在实践方面，它促进了代数曲线理论与密码学中高度相关的应用之间的联系。近年来，椭圆曲线和超椭圆曲线因其在公钥密码学中的重要意义而成为研究的热点。因此，代数几何、数论和算法理论的工具是椭圆和超椭圆曲线密码系统密码分析的核心。该方案的方法可以用来保证这些曲线密码体制的安全性或揭示某些曲线的弱点。这项研究还推进了数论计算，并将各种强有力的最新结果应用于数论的算法方面。

The investigator studies various effective methods in the theory of elliptic, hyperelliptic, and cubic curves and their function fields. Classical problems in these areas are computing class numbers, regulators, and discrete logarithms, as well as determining cardinalities of Jacobians. The first part concerns arithmetical invariants of hyperelliptic and cubic curves. In particular, the investigator and his colleagues hope to advance counting points methods for these curves over large prime fields. Effective methods make use of modular equations, the distribution of the zeroes of the zeta function, the Hasse-Witt matrix, approximation of Euler products, optimized algorithms, and others. The second part concerns with the Weil descent methodology for elliptic curves or other Galois descent methods. The Weil descent methodology is a means to reduce the elliptic curve discrete logarithm problem (ECDLP) over composite finite fields to the discrete logarithm problem in an abelian variety over a proper subfield. This leads to an effective method of reducing any instance of the ECDLP over a finite field to an instance of the discrete logarithm problem in the Jacobian of a hyperelliptic curve over a subfield. Since subexponential-time algorithms for the latter problem are known, this shows how important the method is for cracking certain elliptic curve cryptographic schemes. Similar ideas are applicable for curves of genus bigger than one.The proposed research belongs to the interface between number theory and algebraic geometry. On the theoretical side, it advances the theory of algebraic function fields and curves. At the same time, on the practical side, it advances the connection between the theory of algebraic curves and a highly relevant application to cryptography. In recent years, elliptic and hyperelliptic curves have become objects of intense investigation because of their significance to public-key cryptography. Hereby, tools from algebraic geometry, number theory, and the theory of algorithms are central in the cryptanalysis of elliptic and hyperelliptic curve cryptosystems. Methods of this proposal can be applied to guarantee the security of these curve cryptosystems or reveal weaknesses of certain curves. The proposed research also advances the number theoretic computations and applies a variety of strong recent results to the algorithmic aspects of number theory.