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arXiv:1008.0623 · Physical Review A

Fundamental quantitative security in quantum key generation

Horace P. Yuen

Published Dec 7, 2010• 29 citations
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Abstract

We analyze the fundamental security significance of the quantitative criteria on the final generated key $K$ in quantum key generation including the quantum criterion $d$, the attacker's mutual information on $K$, and the statistical distance between her distribution on $K$ and the uniform distribution. For operational significance a criterion has to produce a guarantee on the attacker's probability of correctly estimating some portions of $K$ from her measurement, in particular her maximum probability of identifying the whole $K$. We distinguish between the raw security of $K$ when the attacker just gets at $K$ before it is used in a cryptographic context and its composition security when the attacker may gain further information during its actual use to help get at $K$. We compare both of these securities of $K$ to those obtainable from conventional key expansion with a symmetric key cipher. It is pointed out that a common belief in the superior security of a quantum generated $K$ is based on an incorrect interpretation of $d$ which cannot be true, and the security significance of $d$ is uncertain. Generally, the quantum key distribution key $K$ has no composition security guarantee and its raw security guarantee from concrete protocols is worse than that of conventional ciphers. Furthermore, for both raw and composition security there is an exponential catch-up problem that would make it difficult to quantitatively improve the security of $K$ in a realistic protocol. Some possible ways to deal with the situation are suggested.

DOI
10.1103/physreva.82.062304

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We analyze the fundamental security significance of the quantitative criteria on the final generated key KKK in quantum key generation including the quantum criterion ddd, the attacker's mutual information on KKK, and the statistical distance between her distribution on KKK and the uniform distribution. For operational significance a criterion has to produce a guarantee on the attacker's probability of correctly estimating some portions of KKK from her measurement, in particular her maximum probability of identifying the whole KKK. We distinguish between the raw security of KKK when the attacker just gets at KKK before it is used in a cryptographic context and its composition security when the attacker may gain further information during its actual use to help get at KKK. We compare both of these securities of KKK to those obtainable from conventional key expansion with a symmetric key cipher. It is pointed out that a common belief in the superior security of a quantum generated KKK is based on an incorrect interpretation of ddd which cannot be true, and the security significance of ddd is uncertain. Generally, the quantum key distribution key KKK has no composition security guarantee and its raw security guarantee from concrete protocols is worse than that of conventional ciphers. Furthermore, for both raw and composition security there is an exponential catch-up problem that would make it difficult to quantitatively improve the security of KKK in a realistic protocol. Some possible ways to deal with the situation are suggested.
The security issues facing quantum key distribution (QKD) are explained, herein focusing on those issues that are cryptographic and information theoretic in nature and not those based on physics. The problem of security criteria is addressed. It is demonstrated that an attacker's success probabilities are the fundamental criteria of security that any theoretic security criterion must relate to in order to have operational significance. The errors committed in the prevalent interpretation of the trace distance criterion are analyzed. The security proofs of QKD protocols are discussed and assessed in regard to three main features: their validity, completeness, and adequacy of the achieved numerical security level. Problems are identified in all these features. It appears that the QKD security situation is quite different from the common perception that a QKD-generated key is nearly perfectly secure. Built into our discussion is a simple but complete quantitative description of the information theoretic security of classical key distribution that is also applicable to the quantum situation. In the Appendixes, we provide a brief outline of the history of some major QKD security proofs, a rather unfavorable comparison of current QKD proven security with that of conventional symmetric key ciphers, and a list of objections and answers concerning some major points of this paper.
The security issues facing quantum key distribution (QKD) are explained, herein focusing on those issues that are cryptographic and information theoretic in nature and not those based on physics. The problem of security criteria is addressed. It is demonstrated that an attacker's success probabilities are the fundamental criteria of security that any theoretic security criterion must relate to in order to have operational significance. The errors committed in the prevalent interpretation of the trace distance criterion are analyzed. The security proofs of QKD protocols are discussed and assessed in regard to three main features: their validity, completeness, and adequacy of the achieved numerical security level. Problems are identified in all these features. It appears that the QKD security situation is quite different from the common perception that a QKD-generated key is nearly perfectly secure. Built into our discussion is a simple but complete quantitative description of the information theoretic security of classical key distribution that is also applicable to the quantum situation. In the appendices, we provide a brief outline of the history of some major QKD security proofs, a rather unfavorable comparison of current QKD proven security with that of conventional symmetric key ciphers, and a list of objections and answers concerning some major points of the paper.
Cryptographic security of quantum key distribution is currently based on a trace distance criterion. The widespread misinterpretation of the criterion as failure probability and also its actual scope have been discussed previously. Recently its distinguishability advantage interpretation is re-emphasized as an operational guarantee, and the failure probability misinterpretation is maintained with a further failure probability per bit interpretation. In this paper we explain the basic perpetuating error as a confusion on the correspondence between mathematics and reality. We note that the assignment of equal a priori probability of 1/2 to the real and ideal situations for distinguishability advantage would not lead to operational guarantee. We explain why operational guarantee in terms of Eve's probabilities of getting various key bits is necessary for security, and why the failure probability interpretation misrepresents the security situation. The scope and limits of the trace distance guarantee are summarized. It is shown that there would have been no security problem to begin with if the failure probability per bit interpretation validity.
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 security significance of the trace distance security criterion ddd is analyzed in terms of operational probabilities of an attacker's success in identifying different subsets of the generated key, both during the key generation process and when the key is used in one-time pad data encryption under known-plaintext attacks. The difference between Eve's sequence error rate and bit error rate is brought out. It is shown with counter-examples that the strong security claim maintained in the literature is incorrect. Other than the whole key error rates that can be quantified at the levels d^{1/3} and d^{1/4} which are much worse than ddd itself, the attacker's success probabilities in estimating various subsets of the key and in known-plaintext attacks are yet to be quantified from ddd if possible. It is demonstrated in realistic numerical examples of concrete protocols that drastic breach of security cannot yet be ruled out.