基于数论函数的新型伪随机数生成器的设计与分析

数论函数是密码学的理论基础之一，像费马小定理、欧拉定理等就直接应用于RSA、ElGamal等公钥密码体制中。本项目将借助费马小定理、欧拉定理等经典性数论性质，进一步探析它们在序列密码中的应用。序列密码的安全性依赖于密钥序列的伪随机性、不可预测性，因此密钥序列的生成是序列密码的关键技术。我们将以费马小定理、欧拉定理相关的数论函数--费马商、欧拉商和Carmichael商为工具，一方面进一步讨论其数学解析性质、其与离散对数的内在联系，提出新的商式函数；另一方面应用它们设计新型伪随机数（序列）生成算法，以构造适合密码学应用的安全的新型伪随机数（序列），结合有限域（环）理论、指数和理论深入分析序列的分布、相关性、（非）线性复杂度、k-错复杂度等密码学指标；并将相关方法进行延拓，探索p-adic域上费马商的应用。项目的研究将为序列密码提供新的密码算法及性能良好的密钥序列，进一步完善密码理论。

Arithmetical functions are one of theoritical foundations of cryptography, including Fermat Little Theorem and Euler Theorem, which are used directly in RSA and ElGamal public cryptosystems. the project mainly aims at the applications of Fermat Little Theorem and Euler Theorem to stream ciphers, whose security heavily depends on the pseudorandomness and predictability of key-streams. Thus the generation of key-streams is a key technique of stream ciphers. We will apply Fermat quotients, Euler quotients and Carmichael quotients, which are related to Fermat Little Theorem and Euler Theorem, to designing new algorithms for generating pseudorandom sequences with security cryptographic properties. On one hand, we will discuss the mathematical properties, study a relationship between Fermat quotients, Euler quotients and Carmichael quotients and discrete logarithm, discribe new quotients. On the other hand, we investigate cryptographic measures of the sequences, such as distribution, correlation, linear complexity, non-linear complexity and k-error linear complexity using the theory of finite fields (rings) and exponential sums. At the same time, we will extend our methods to Fermat quotients over p-adic fields and try to discuss some cryptographic applications. The research will present new aoglrithms of key-stream sequences with good cryptographic properties and improve the theory of stream ciphers furtherly.