EKE Meets Tight Security in the Universally Composable Framework

Abstract

(Asymmetric) Password-based Authenticated Key Exchange ((a)PAKE) protocols allow two parties establish a session key with a pre-shared low-entropy password. In this paper, we show how Encrypted Key Exchange (EKE) compiler [Bellovin and Merritt, S &P 1992] meets tight security in the Universally Composable (UC) framework. We propose a strong 2DH variant of EKE, denoted by 2DH-EKE, and prove its tight security in the UC framework based on the CDH assumption. The efficiency of 2DH-EKE is comparable to the original EKE, with only O(λ) bits growth in communication (λ the security parameter), and two (resp., one) extra exponentiation in computation for client (resp., server). We also develop an asymmetric PAKE scheme 2DH-aEKE from 2DH-EKE. The security reduction loss of 2DH-aEKE is N, the total number of client-server pairs. With a meta-reduction, we formally prove that such a factor N is inevitable in aPAKE. Namely, our 2DH-aEKE meets the optimal security loss. As a byproduct, we further apply our technique to PAKE protocols like SPAKE2 and PPK in the relaxed UC framework, resulting in their 2DH variants with tight security from the CDH assumption.