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.
