Number-theoretic cryptography

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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.