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arXiv:1205.3820

Problems of Security Proofs and Fundamental Limit on Key Generation Rate in Quantum Key Distribution

Horace P. Yuen

Published May 16, 2012• 11 citations
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Abstract

It is pointed out that treatments of the error correcting code in current quantum key distribution protocols of the BB84 type are not correct under joint attack, and the general interpretation of the trace distance security criterion is also incorrect. With correct interpretation of the criterion as well as a correct treatment of the error correcting code and privacy amplification code, it is shown that even for an ideal system under just collective attack, the maximum tolerable quantum bit error rate is about 1.5% and a net key cannot actually be generated with practical error correcting codes even at such low rates, contrary to claims in the literature.

DOI
10.48550/arxiv.1205.3820

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It is pointed out that treatments of the error correcting code in current quantum key distribution protocols of the BB84 type are not correct under joint attack, and the general interpretation of the trace distance security criterion is also incorrect. With correct interpretation of the criterion as well as a correct treatment of the error correcting code and privacy amplification code, it is shown that even for an ideal system under just collective attack, the maximum tolerable quantum bit error rate is about 1.5% and a net key cannot actually be generated with practical error correcting codes even at such low rates, contrary to claims in the literature.
Privacy amplification is a necessary step in all quantum key distribution protocols, and error correction is needed in each except when signals of many photons are used in the key communication in quantum noise approach. No security analysis of error correcting code information leak to the attacker has ever been provided, while an ad hoc formula is currently employed to account for such leak in the key generation rate. It is also commonly believed that privacy amplification allows the users to at least establish a short key of arbitrarily close to perfect security. In this paper we show how the lack of rigorous error correction analysis makes the otherwise valid privacy amplification results invalid, and that there exists a limit on how close to perfect a generated key can be obtained from privacy amplification. In addition, there is a necessary tradeoff between key rate and security, and the best theoretical values from current theories would not generate enough near-uniform key bits to cover the message authentication key cost in disturbance-information tradeoff protocols of the BB84 variety.
It has been widely claimed and believed that many protocols in quantum key distribution, especially the single-photon BB84 protocol, have been proved unconditionally secure at least in principle, for both asymptotic and finite protocols with realistic bit lengths. In this paper it is pointed out that the only known quantitative justification for such claims is based on incorrect assertions. The precise security requirements are described in terms of the attacker's sequence and bit error probabilities in estimating the key. The extent to which such requirements can be met from a proper trace distance criterion is established. The results show that the quantitative security levels obtainable in concrete protocols with ideal devices do not rule out drastic breach of security unless privacy amplification is more properly applied.
All the currently available unconditional security proofs on quantum key distribution, in particular for the BB84 protocol and its variants including continuous-variable ones, are invalid or incomplete at many points. In this paper we discuss some of the main known problems, particularly those on operational security guarantee and error correction. Most basic are the points that there is no security parameter in such protocols and it is not the case the generated key is perfect with probability ≥1−ε\geq 1-ε≥1−ε under the trace distance criterion d≤εd\leqεd≤ε, which is widely claimed in the technical and popular literature. The many serious security consequences of this error about the QKD generated key would be explained, including practical ramification on achievable security levels. It will be shown how the error correction problem alone may already defy rigorous quantitative analysis. Various other problems would be touched upon. It is pointed out that rigorous security guarantee of much more efficient quantum cryptosystems may be obtained by abandoning the disturbance-information tradeoff principle and utilizing instead the known KCQ (keyed communication in quantum noise) principle in conjunction with a new DBM (decoy bits method) principle that will be detailed elsewhere.
All the currently available unconditional security proofs on quantum key distribution, in particular for the BB84 protocol and its variants including continuous-variable ones, are invalid or incomplete at many points. In this paper we discuss some of the main known problems, particularly those on operational security guarantee and error correction. Most basic are the points that there is no security parameter in such protocols and it is not the case the generated key is perfect with probability ≥ 1 - ϵ under the trace distance criterion d ≤ ϵ, which is widely claimed in the technical and popular literature. The many serious security consequences of this error about the QKD generated key would be explained, including practical ramification on achievable security levels. It will be shown how the error correction problem alone may already defy rigorous quantitative analysis. Various other problems would be touched upon. It is pointed out that rigorous security guarantee of much more efficient quantum cryptosystems may be obtained by abandoning the disturbance-information tradeoff principle and utilizing instead the known KCQ (keyed communication in quantum noise) principle in conjunction with a new DBM (decoy bits method) principle that will be detailed elsewhere.
It is repeatedly and persistently claimed in the literature that a specific trace criterion ddd would guarantee universal composition security in quantum cryptography. Currently that is the sole basis of unconditional security claim in quantum key distribution. In this paper, it is shown that just security against known-plaintext attacks when the generated key is used in direct encryption is not provided by ddd. The problem is directly connected with several general problems in the existing unconditional security proofs in quantum key distribution. A number of issues will be clarified concerning the nature of true security, privacy amplification, key generation rate and the mathematical approach needed for their determination in concrete protocols.
The trace distance criterion for security of Quantum Key Distribution (QKD) has been widely perceived that its upperbound is the maximum failure probability in distributing an ideal quantum state that the quantum system shared by legitimate users is decoupled from the quantum system the eavesdropper possesses, therefore the eavesdropper would not obtain any hints on the key shared by the legitimate users no matter how much her measurement is optimal. However, there an arbitrariness in the definition of the trace distance with the decoupled quantum system the eavesdropper possesses as the previous work pointed out, and such an arbitrariness would be confusions in guaranteeing the security of QKD. In this work, from the framework of Bit-Error-Rate (BER) Guarantee discussed also in the previous work, such an arbitrariness is removed from the definition of the trace distance criterion by discussing the similarities and differences between the security of QKD and classical information-theoretic cryptography by Leftover Hash Lemma (LHL). However, in the previous work BER Guarantee was derived under the assumption that the eavesdropper launches the weakest class of attack, so-called “Individual Attacks.” Therefore, it is still unknown whether direct upper-bounding of the term would give tighter upper-bound in general, such as “Collective Attacks” or “Coherent Attacks”, compared to the upper-bound by LHL, as well as the secure-key generation rate. This study also revisits the security proof of QKD given by P. Shor and J. Preskill in 2000.