The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, a strictly more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system is shown. Our result applies to almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.
Recently, the problem of randomness extraction with quantum side information has generated renewed interest. It has been shown that the classical technique @cite_20 of XORing a classical source about which an adversary holds quantum information with a @math -biased mask results in a uniformly distributed string @cite_4 See also @cite_2 for a generalization of this work to the fully quantum setting. .
@cite_4 @cite_20 @cite_2
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Randomness extraction is of fundamental importance for information-theoretic cryptography. It allows to transform a raw key about which an attacker has some limited knowledge into a fully secure random key, on which the attacker has essentially no information. Up to date, only very few randomness-extraction techniques are known to work against an attacker holding quantum information on the raw key. This is very much in contrast to the classical (non-quantum) setting, which is much better understood and for which a vast amount of different techniques are known and proven to work. We prove a new randomness-extraction technique, which is known to work in the classical setting, to be secure against a quantum attacker as well. Randomness extraction is done by xor'ing a so-called δ-biased mask to the raw key. Our result allows to extend the classical applications of this extractor to the quantum setting. We discuss the following two applications. We show how to encrypt a long message with a short key, information-theoretically secure against a quantum attacker, provided that the attacker has enough quantum uncertainty on the message. This generalizes the concept of entropically-secure encryption to the case of a quantum attacker. As second application, we show how to do errorcorrection without leaking partial information to a quantum attacker. Such a technique is useful in settings where the raw key may contain errors, since standard error-correction techniques may provide the attacker with information on, say, a secret key that was used to obtain the raw key. In the classical privacy amplification problem Alice and Bob share information that is only partially secret towards an eavesdropper Charlie. Their goal is to distill this information to a shorter string that is completely secret. The classical privacy amplification problem can be solved almost optimally using extractors. An interesting variant of the problem, where the eavesdropper Charlie is allowed to keep quantum information rather than just classical information, was introduced by Konig, Maurer and Renner. In this setting, the eavesdropper Charlie may entangle himself with the input (without changing it) and the only limitation Charlie has is that it may keep at most b qubits of storage. A natural question is whether there are classical extractors that are good even against quantum storage. Recent work has shown that some classical extractors miserably fail against quantum storage. At the same time, it was shown that some other classical extractors work well even against quantum storage, but all these extractors had a large seed length that was either as large as the extractor output, or as large as the quantum storage available to the eavesdropper. In this paper we show that a modified version of Trevisan's extractor is good even against quantum storage, thereby giving the first such construction with logarithmic seed length. The technique we use is a combination of Trevisan's approach of constructing an extractor from a black-box pseudorandom generator, together with locally list-decodable codes and previous work done on quantum random access codes. We address the question whether quantum memory is more powerful than classical memory. In particular, we consider a setting where information about a random n-bit string X is stored in s classical or quantum bits, for s<n, i.e., the stored information is bound to be only partial. Later, a randomly chosen predicate F about X has to be guessed using only the stored information. The maximum probability of correctly guessing F(X) is then compared for the cases where the storage device is classical or quantum mechanical, respectively. We show that, despite the fact that the measurement of quantum bits can depend arbitrarily on the predicate F, the quantum advantage is negligible already for small values of the difference n-s. Our setting generalizes the setting of who considered the problem of guessing an arbitrary bit (i.e., one of the n bits) of X. An implication for cryptography is that privacy amplification by universal hashing remains essentially equally secure when the adversary's memory is allowed to be quantum rather than only classical. Since privacy amplification is a main ingredient of many quantum key distribution (QKD) protocols, our result can be used to prove the security of QKD in a generic way. We give an exponential separation between one-way quantum and classical communication protocols for twopartial Boolean functions, both of which are variants of the Boolean Hidden Matching Problem of Bar- Earlier such an exponential separation was known only for a relational version of the Hidden Matching Problem. Our proofs use the Fourier coefficients inequality of Kahn, Kalai, and Linial. We give a number of applications of this separation. In particular, in the bounded-storage model of cryptography we exhibita scheme that is secure against adversaries with a certain amount of classical storage, but insecure against adversaries with a similar (or even much smaller) amount of quantum storage; in the setting of privacy amplification, we show that there are strong extractors that yield a classically secure key, but are insecure against a quantum adversary.