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arXiv:quant-ph/0108013

Privacy Amplification in Quantum Key Distribution: Pointwise Bound versus Average Bound

Gerald N. Gilbert, Michael Hamrick, F. Javier Thayer

Published Aug 2, 2001• 3 citations

Abstract

In order to be practically useful, quantum cryptography must not only provide a guarantee of secrecy, but it must provide this guarantee with a useful, sufficiently large throughput value. The standard result of generalized privacy amplification yields an upper bound only on the average value of the mutual information available to an eavesdropper. Unfortunately this result by itself is inadequate for cryptographic applications. A naive application of the standard result leads one to incorrectly conclude that an acceptable upper bound on the mutual information has been achieved. It is the pointwise value of the bound on the mutual information, associated with the use of some specific hash function, that corresponds to actual implementations. We provide a fully rigorous mathematical derivation that shows how to obtain a cryptographically acceptable upper bound on the actual, pointwise value of the mutual information. Unlike the bound on the average mutual information, the value of the upper bound on the pointwise mutual information and the number of bits by which the secret key is compressed are specified by two different parameters, and the actual realization of the bound in the pointwise case is necessarily associated with a specific failure probability. The constraints amongst these parameters, and the effect of their values on the system throughput, have not been previously analyzed. We show that the necessary shortening of the key dictated by the cryptographically correct, pointwise bound, can still produce viable throughput rates that will be useful in practice.

DOI: 10.48550/arxiv.quant-ph/0108013

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We study the practical effectiveness of privacy amplification for classical key distribution schemes. We find that in contrast to quantum key distribution schemes, the high fidelity of the raw key generated in classical systems allow the users to always sift a secure shorter key if they have an upper bound on the eavesdropper probability to correctly guess the exchanged key bits. The number of privacy amplification iterations needed to achieve information leak of 10−8 in existing classical communicators is 2 or 3 resulting in a corresponding slowdown 4 to 8. We analyze the inherent tradeoff between the number of iterations and the security of the raw key. This property which is unique to classical key distribution systems render them highly useful for practical, especially for noisy channels where sufficiently low quantum bit error ratios are difficult to achieve.

We study the practical effectiveness of privacy amplification for classical key distribution schemes. We find that in contrast to quantum key distribution schemes, the high fidelity of the raw key generated in classical systems allow the users to always sift a secure shorter key if they have an upper bound on the eavesdropper probability to correctly guess the exchanged key bits. The number of privacy amplification iterations needed to achieve information leak of 10 −8 in existing classical communicators is 2 or 3 resulting in a corresponding slowdown 4 to 8. We analyze the inherent tradeoff between the number of iterations and the security of the raw key. This property which is unique to classical key distribution systems render them highly useful for practical, especially for noisy channels where sufficiently low quantum bit error ratios are difficult to achieve.

We analyse the information obtained by an eavesdropper during the various stages of a quantum cryptographic protocol associated with key distribution. We provide both an upper and a lower limit on the amount of information that may have leaked to the eavesdropper at the end of the key distribution procedure. These limits are restricted to intercept/resend eaves-dropping strategies. The upper one is higher than has been estimated so far, and should be taken into account in order to guarantee the secrecy of the final key, which is subsequently obtained via the so-called privacy amplification.

Quantum cryptography has attracted much recent attention due to its potential for providing secret communications that cannot be decrypted by any amount of computational effort. This is the first analysis of the secrecy of a practical implementation of the BB84 protocol that simultaneously takes into account and presents the full set of analytical expressions for effects due to the presence of pulses containing multiple photons in the attenuated output of the laser, the finite length of individual blocks of key material, losses due to error correction, privacy amplification, and authentication, errors in polarization detection, the efficiency of the detectors, and attenuation processes in the transmission medium. The analysis addresses eavesdropping attacks on individual photons rather than collective attacks in general. Of particular importance is the first derivation of the necessary and sufficient amount of privacy amplification compression to ensure secrecy against the loss of key material which occurs when an eavesdropper makes optimized individual attacks on pulses containing multiple photons. It is shown that only a fraction of the information in the multiple photon pulses is actually lost to the eavesdropper.

This paper examines classical privacy amplification using a universal family of hash functions. In quantum key distribution, the adversary's measurement can wait until the choice of hash functions is announced, and so the adversary's information may depend on the choice. Therefore the existing result on classical privacy amplification, which assumes the independence of the choice from the other random variables, is not applicable to this case. This paper provides a security proof of privacy amplification which is valid even when the adversary's information may depend on the choice of hash functions. The compression rate of the proposed privacy amplification can be taken to be the same as that of the existing one with an exponentially small loss in secrecy of a final key.

Quantum Key Distribution (QKD) protocols typically involve two separate steps of error-verification and privacy amplification. Both steps involve the application of a hash function sampled from a suitable hash family to classical data. A potentially simpler alternative is to implement a single-step hashing procedure, announce and compare a fixed number of bits of its output for error-verification, and retain the remaining bits as the final output key. This reduces the complexity of classical postprocessing, the amount of classical communication, and the amount of randomness needed in the protocol. While this construction has appeared in the literature, a suitable mathematical justification is lacking. In this work, we show that the security of such protocols, under certain simple conditions, follows from a slightly modified notion of secrecy of the protocol without the final error-verification step. In particular, we show that for most entropy-based proofs, the combined protocol can be implemented with no change to the key rate, and a minor change to the security parameter.