Number-theoretic cryptography

公钥密码系统是私人、商业和政府用户广泛采用的一项关键技术，用于确保电子数据通信的隐私性和真实性。大多数公钥加密系统的安全性是基于某个数论问题的难易程度。这些问题被认为是困难的，但没有被证明。最常见的例子是整数分解问题，它构成了RSA （Rivest-Shamir-Adleman）密码系统的基础。与此建议相关的例子是椭圆曲线上点群的离散对数问题（ECDLP）和有限域上多元二次方程（MQ）方程组的求解问题。本研究旨在进一步探讨所选公钥密码体制的数论和代数性质。这包括与安全相关的方面和效率方面。具体来说，我们希望进一步确定ECDLP可能不太安全的实例。考虑到基于椭圆曲线的密码系统在受限的计算环境（如智能卡）中代表了最有效的安全增强解决方案，这些结果对研究人员和从业人员都非常感兴趣。我们进一步期望使用双线性对为密码系统的高安全性应用生成最优系统参数。在密码学中使用双线性对已经导致了令人兴奋的新应用，例如第一个可行且可证明安全的基于身份的加密方案。此外，我们期望更好地理解基于MQ方程的密码系统的安全性。这样的密码系统还有一个额外的优势，那就是它们不容易受到量子计算机的攻击。在我们的研究中，我们将使用计算数论和代数几何的工具，结合实验工作和理论研究。

Public-key cryptosystems are a key technology widely deployed by private, commercial and governmental users to ensure privacy and authenticity in electronic data communications. The security of most public-key cryptographic systems is based on the difficulty of a certain number-theoretic problem. These problems are believed, but not proven, to be hard. The most popular example is the integer factorization problem, which forms the basis for the Rivest-Shamir-Adleman (RSA) cryptosystem. Examples relevant for this proposal are the discrete logarithm problem in the group of points on an elliptic curve (ECDLP) and the problem of finding solutions to systems of multivariate quadratic (MQ) equations over finite fields. The proposed research is to further explore the number-theoretic and algebraic properties of selected public-key cryptosystems. This includes both security-related aspects and aspects of efficiency.   Specifically, we expect to identify further possibly less secure instances of the ECDLP. Such results are of extreme interest to both researchers and practitioners, given that elliptic curve-based cryptosystems represent the most efficient security-enhancing solution in constrained computing environments such as smart cards. We further expect to generate optimal system parameters for high-security applications of cryptosystems using bilinear pairings. The use of bilinear pairings in cryptography has led to exciting novel applications such as the first workable and provably secure identity-based encryption scheme. Moreover, we expect to achieve a better understanding of the security of cryptosystems based on MQ equations. Such cryptosystems have the added advantage that they are not vulnerable to attacks performed by quantum computers. In our research, we will be using tools from computational number theory and algebraic geometry, combining experimental work and theoretical investigation.