International Association
for Cryptologic Research

We revisit one of the most fundamental hardness amplification constructions, originally proposed by Yao (FOCS 1982). We present a hardness amplification theorem for the direct product of certain games that is simpler, more general, and stronger than previously known hardness amplification theorems of the same kind. Our focus is two-fold. First, we aim to provide close-to-optimal concrete bounds, as opposed to asymptotic ones. Second, in the spirit of abstraction and reusability, our goal is to capture the essence of direct product hardness amplification as generally as possible. Furthermore, we demonstrate how our amplification theorem can be applied to obtain hardness amplification results for non-trivial interactive cryptographic games such as MAC forgery or signature forgery games.

This paper makes three contributions. First, we present a simple theory of random systems. The main idea is to think of a probabilistic system as an equivalence class of distributions over deterministic systems. Second, we demonstrate how in this new theory, the optimal information-theoretic distinguishing advantage between two systems can be characterized merely in terms of the statistical distance of probability distributions, providing a more elementary understanding of the distance of systems. In particular, two systems that are epsilon-close in terms of the best distinguishing advantage can be understood as being equal with probability 1-epsilon, a property that holds statically, without even considering a distinguisher, let alone its interaction with the systems. Finally, we exploit this new characterization of the distinguishing advantage to prove that any threshold combiner is an amplifier for indistinguishability in the information-theoretic setting, generalizing and simplifying results from Maurer, Pietrzak, and Renner (CRYPTO 2007).