There are often situations where two remote users each have data, and wish to (i) verify the equality of their data, and (ii) whenever a discrepancy is found afterwards, determine which of the two modified his data. The most common example is where they want to authenticate messages they exchange. Another possible example is where they have a huge database and its mirror in remote places, and whenever a discrepancy is found between their data, they can determine which of the two users is to blame.Of course, if one is allowed to use computational assumptions, this function can be realized readily, e.g., by using digital signatures. However, if one needs information-theoretic security, there is no known method that realizes this function efficiently, i.e., with secret key, communication, and trusted third parties all being sufficiently small.In order to realize this function efficiently with information-theoretic security, we here define the "equality-testing protocol with dispute resolution" as a new framework. The most significant difference between our protocol and the previous methods with similar functions is that we allow the intervention of a trusted third party when checking the equality of the data. In this new framework, we also present an explicit protocol that is information-theoretically secure and efficient.

Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.

We show that the Mayers-Shor-Preskill approach and Renner's approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD.

We explicitly construct random hash functions for privacy amplification (extractors) that require smaller random seed lengths than the previous literature, and still allow efficient implementations with complexity O(n log n) for input length n. The key idea is the concept of dual universal2hash function introduced recently. We also use a new method for constructing extractors by concatenating δ-almost dual universal2hash functions with other extractors. Besides minimizing seed lengths, we also introduce methods that allow one to use non-uniform random seeds for extractors. These methods can be applied to a wide class of extractors, including dual universal2hash function, as well as to the conventional universal2hash functions.

We explicitly construct random hash functions for privacy amplification (extractors) that require smaller random seed lengths than the previous literature, and still allow efficient implementations with complexity O(n log n) for input length n. Firstly, we construct two types of hash functions by using the finite-filed. Then, concatenating them, we construct other two types of hash functions. We compare our hash functions with existing hash function in an asymptotic setting under a fixed key generation rate.

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 familyFand a code familyC, and thereby defining ε-almost dual universal2hash functions, as a generalization of the conventional universal2hash 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 δ-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. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane (CSS) type, this result removes the existing difficulty of the construction of a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. 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 δ-biased family.