Dual universality of hash functions and its applications to classical and quantum cryptography

In this paper, we introduce the concept of dual universality of hash functions and present its applications to various quantum and classical communication models including cryptography. We begin by establishing the one-to-one correspondence between a linear function family F and a code family C, and thereby defining ε-almost dual universal2 hash functions, as a generalization of the conventional universal2 hash functions. Then we give a security proof for the Bennett-Brassard 1984 protocol, where the Shor-Preskill–type argument is used, but nevertheless ε-almost dual universal2 functions can be used for privacy amplification. We show that a similar result applies to the quantum wire-tap channel as well. We also apply these results on quantum models for investigating the classical wire-tap channel and randomness extraction, and obtain various new results, including the existence of a deterministic hash function that is universally secure against different types of wire-tapper. For proving these results, we present an extremely simple argument by simulating the classical channels by quantum channels, where the strength of Eve’s wire-tapping can be measured by the phase bit error rate. Under this setting of quantum simulation, we demonstrate that our ε-almost dual universal2 functions are more relevant for security than the conventional ε-almost universal2 hash functions, by showing that the former functions correspond to a linear code family having an appropriate phase-error correcting property. These examples suggest the importance of quantum approaches in classical settings of information theory, as well as the dual universality of hash functions.