The efficiency and security of a real quadratic field based key exchange protocol

Abstract

Most cryptographic key exchange protocols make use of the presumed difficulty of solving the discrete logarithm problem (DLP) in a certain finite group as the basis of their security. Recently, real quadratic number fields have been proposed for use in the development of such protocols. Breaking such schemes is known to be at least as difficult a problem as integer factorization; furthermore, these are the first discrete logarithm based systems to utilize a structure which is not a group, specifically the collection of reduced ideals which belong to the principal class of the number field. For this structure the DLP is essentially that of determining a generator of a given principal ideal. Unfortunately, there are a few implementation-related disadvantages to these schemes, such as the need for high precision floating point arithmetic and an ambiguity problem that requires a short, second round of communication. In this paper we describe work that has led to the resolution of some of these difficulties. Furthermore, we discuss the security of the system, concentrating on the most recent techniques for solving the DLP in a real quadratic number field.