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Journal of Experimental and Theoretical Physics

One-time pad, complexity of verification of keys, and practical security of quantum cryptography

S. N. Molotkov

Published Nov 1, 2016• 10 citations
Publisher

Abstract

A direct relation between the complexity of the complete verification of keys, which is one of the main criteria of security in classical systems, and a trace distance used in quantum cryptography is demonstrated. Bounds for the minimum and maximum numbers of verification steps required to determine the actual key are obtained.
DOI
10.1134/s1063776116110169

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The trace distance is used as a security criterion in proofs of security of keys in quantum cryptography. Some authors doubted that this criterion can be reduced to criteria used in classical cryptography. The following question has been answered in this work. Let a quantum cryptography system provide an ε-secure key such that ½‖ρ XE − ρ U ⊗ ρ E ‖1 < ε, which will be repeatedly used in classical encryption algorithms. To what extent does the ε-secure key reduce the number of search steps (guesswork) as compared to the use of ideal keys? A direct relation has been demonstrated between the complexity of the complete consideration of keys, which is one of the main security criteria in classical systems, and the trace distance used in quantum cryptography. Bounds for the minimum and maximum numbers of search steps for the determination of the actual key have been presented.
The proof of the security of quantum key distribution is a rather complex problem. Security is defined in terms different from the requirements imposed on keys in classical cryptography. In quantum cryptography, the security of keys is expressed in terms of the closeness of the quantum state of an eavesdropper after key distribution to an ideal quantum state that is uncorrelated to the key of legitimate users. A metric of closeness between two quantum states is given by the trace metric. In classical cryptography, the security of keys is understood in terms of, say, the complexity of key search in the presence of side information. In quantum cryptography, side information for the eavesdropper is given by the whole volume of information on keys obtained from both quantum and classical channels. The fact that the mathematical apparatuses used in the proof of key security in classical and quantum cryptography are essentially different leads to misunderstanding and emotional discussions [1]. Therefore, one should be able to answer the question of how different cryptographic robustness criteria are related to each other. In the present study, it is shown that there is a direct relationship between the security criterion in quantum cryptography, which is based on the trace distance determining the distinguishability of quantum states, and the criterion in classical cryptography, which uses guesswork on the determination of a key in the presence of side information.
Abstract We discuss the operational meaning of a commonly accepted security parameter in quantum key distribution, which is based on the trace distance. We separately consider the cases of using a key in a one-time pad and in a computationally secure cipher. Some practical aspects of using the security parameter are also elucidated, which are usually not paid enough attention in theoretical studies and which therefore may cause difficulties for experimentalists and engineers. It is shown that a one-time pad requires not only a higher key generation rate than computationally secure ciphers, but also a significantly stronger condition on the key security parameter.
Abstract In practice today, quantum keys are often reused because of practical limitations. It is clear that reusing a key weakens security compared to one-time-pad encryption, where perfect secrecy is achieved. For this reason, it is important to understand how the effort needed to find the key changes when the key is reused. We use the trace distance between two quantum states as a measure of closeness. Its maximum value is one, which means the two states are as different as possible. With simple arguments, we show that the trace distance is bounded above by several terms that capture (i) non-ideal quantum keys, (ii) plaintext redundancy, (iii) the number of key reuses, and (iv) the number of collisions of the block cipher. The trace distance approaches one exponentially fast with plaintext redundancy and with the number of key reuses; this means brute-force search becomes as easy as possible. For a random block cipher, the expected trace distance grows linearly with the number of key uses, with coefficient e −1 ( e is Euler’s constant). The result is information-theoretic in nature.
\n\t\t\t\t\tMost of cryptographic techniques today provide secure enough communications in society. However, none of them can provide unconditional security (except One-Time pad, which is the only such mechanism). They all have weaknesses. That is why they should be consistently updated and renovated along with changes in technology. However, the quantum computers have potential to completely revolutionize cryptography. They would be so powerful that all complex computations and mathematical problems on which current cryptography rely, would become trivial. They would be able to solve them in a matter of seconds. At the same time quantum cryptography would offer perfectly secure communication. This paper is brief introduction to quantum computing, quantum cryptography, and to existing quantum technology. \n\t\t\t\t
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.