Metaplectic Forms and Cryptography

William D.本研究将集中在数论的两个不同领域。 首先，研究者将继续研究元形式的Whittaker-Fourier系数，以期在数论中寻求新的算术和分析应用，例如改进的附加于n阶Hecke特征标的L-级数的平均值结果，以及GL（n）上L-函数的Bump-Hoffstein公式的推广。 其次，研究人员和他的同事将继续在密码学的一些领域进行联合研究，包括开发有效的识别方案，新的加密技术，以及伪随机函数理论的进一步成果。自守形式（例如与椭圆曲线相关的自守形式）在现代数论中扮演着重要和核心的角色，允许复分析的强大工具来解决算术问题。 亚辛形式是自守形式的自然（虽然更神秘）推广，其构造密切依赖于类场论的非常深刻的结果（特别是高阶互反律）。 虽然元形式的研究仍处于起步阶段，但这项研究的结果将被证明是非常令人兴奋的。 本提案的第二部分侧重于密码学，这是一个快速发展的数学/计算机科学领域，主要涉及信息安全。 研究人员和他的同事们计划继续他们的工作，识别方案使用信用卡/智能卡，加密算法，以及与伪随机函数的理论有关的问题。

William D. Banks This research will focus on two somewhat distinct areas of number theory. First, the investigator will continue to study the Whittaker-Fourier coefficients of metaplectic forms, with a view to seeking new arithmetical and analytical applications in number theory, such as improved average value results for L-series attached to n-th order Hecke characters, and generalizations of the Bump-Hoffstein conjectures for L-functions on GL(n). Second, the investigator and his colleagues will continue their joint research in number of areas of cryptography, including the development of efficient identification schemes, new techniques of encryption, and further results in the theory of pseudorandom functions. Automorphic forms (such as those associated to elliptic curves) play an important and central role in modern number theory, allowing the powerful tools of complex analysis to come to bear on arithmetical questions. Metaplectic forms are a natural (though more mysterious) generalization of automorphic forms whose construction intimately depends on very deep results of class field theory (in particular, the higher order reciprocity laws). While the study of metaplectic forms in still in its infancy, the results of this proposed research will prove to be very exciting. The second part of this proposal focuses on cryptography, the rapidly developing field of mathematics/computer science that deals primarily with information security. The investigator and his colleagues plan to continue their work on identification schemes for use with credit/smart cards, encryption algorithms, and questions related to the theory of pseudorandom functions.