Dual universality of hash functions and its applications to quantum cryptography

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

2010-12-30

arxiv.org/abs/1101.0064v3

Physics

Quantum Physics

16 pages, 2 figures; revised argument concerning the relation with the \delta-biased family

Scientific paper

In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function family {\cal F} and a code family {\cal C}, and thereby defining \epsilon-almost dual universal_2 hash functions, as a generalization of the conventional universal_2 hash functions. Then we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the \delta-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. This result removes the existing difficulty of the Shor-Preskill formalism that it requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane type. We also show that a similar result applies to the quantum wire-tap channel. Finally we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill--type argument gives better security bounds in terms of the trace distance and Holevo information, than the method based on the \delta-biased family.

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