Effective Methods for Hyperelliptic and Cubic Function Fields

研究人员研究椭圆，超椭圆和三次曲线及其函数域理论中的各种有效方法。在这些领域的经典问题是计算类数，调节器，和离散方程，以及确定雅可比基数。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，研究人员和他的同事们希望在大型素数场上推进这些曲线的计数点方法。有效的方法利用模方程，zeta函数的零点分布，Hasse-Witt矩阵，欧拉乘积的近似，优化算法等。第二部分涉及椭圆曲线的Weil下降方法或其他Galois下降方法。Weil下降法是将复合有限域上的椭圆曲线离散对数问题（ECDLP）转化为一个适当子域上的阿贝尔簇上的离散对数问题的一种方法。这导致了一个有效的方法，减少任何实例的ECDLP在有限域的一个例子的离散对数问题的雅可比矩阵的超椭圆曲线在一个子域。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1的曲线，本文的研究属于数论与代数几何的结合。在理论方面，提出了代数函数域和代数函数曲线的理论。与此同时，在实践方面，它推进了代数曲线理论与密码学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由于其在公钥密码学中的重要性而成为研究的热点。因此，工具从代数几何，数论和理论的算法是中央的密码分析的椭圆和超椭圆曲线密码系统。该方案的方法可用于保证这些曲线密码系统的安全性或揭示某些曲线的弱点。拟议的研究还提出了数论计算和应用各种强大的最近的结果数论的算法方面。

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.