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Laser Physics Letters

Shannon unicity distance, ε -secrecy criterion and Holevo information in quantum cryptography

Igor M. Arbekov, S. N. Molotkov

Published Dec 21, 2020• 0 citations
Publisher

Abstract

Abstract In this paper, we consider the relationship between C Shannon’s unicity distance, which is determined by the number of symbols of the encrypted message required to uniquely find the encryption key of the cryptosystem in information-theoretical sense, ε -secrecy criterion in quantum cryptography and Holevo information. A lower bound for the unicity distance for ε -secret keys in quantum cryptography is obtained. The relationship between unicity distance, ε -secrecy criterion in quantum cryptography and Holevo information is demonstrated.

DOI
10.1088/1612-202x/abd173

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Abstract In this paper, we consider the relationship between C Shannon’s unicity distance, which is determined by the number of symbols of the encrypted message required to uniquely find the encryption key of the cryptosystem in information-theoretical sense, ε -secrecy criterion in quantum cryptography and Holevo information. A lower bound for the unicity distance for ε -secret keys in quantum cryptography is obtained. The relationship between unicity distance, ε -secrecy criterion in quantum cryptography and Holevo information is demonstrated.
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 Basic configuration in quantum cryptography is a point-to-point quantum key distribution. In networks with quantum key distribution, the keys received on different network segments are different. For information encryption between arbitrary network nodes it is necessary to have agreement of the keys distributed in different segments of network through a public classical communication channel. The secrecy criterion for keys independently distributed in different network segments is formulated in terms of the trace distance. These are the so-called ε –secret keys. The paper deals with the issue of the secrecy of the agreed keys. We propose the method of secret keys agreement in which the agreement of two ε 1 , ε 2 –secret keys distributed in different segments leads to a common ( ε 1 + ε 2 ) –secret key. It is shown that in order to agreed common key will be ε –secret, it is sufficient that the keys on the individual segments are ε /2–secret.