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International Journal of Modern Physics B

QUANTUM KEY DISTRIBUTION WITH SEVERAL INTERCEPTS AND RESEND ATTACKS

Hamid Ez-zahraouy A, A. Benyoussef

Published Sep 20, 2009• 10 citations

Abstract

The effect of several eavesdroppers intercept-resend attacks on the quantum error and mutual information between honest parties of a quantum channel chain is investigated within the BB84 (Bennett and Brassard, 1984). The quantum error and the mutual information are computed for arbitrary number of attacks. It is found that the quantum error and the secured–no secured transition depend strongly on the number N of eavesdroppers and their probabilities of intercepting attacks. For N = 3, numerical calculations show that the quantum error exhibits three kinds of different behaviors as a function of the probability of attack ω1of the first eavesdropper namely: (i) the quantum error remains constant for sufficiently small ω1, (ii) exhibits a plateau for intermediate values of ω1, (iii) increases, passes through a maximum and decreases when increasing ω1. However, depending on the probabilities of attack, phase diagrams present several kinds of topologies, in good agreement with the quantum error behavior. Moreover, in the particular case where all eavesdroppers intercept with identical probabilities ω, the quantum error increases as a nonlinear function with the number of eavesdroppers before reaching an upper limit of ≈ 0.475 for sufficiently large N. Besides, it is shown that the secured–no secured transition occurs under the effect of the number of eavesdroppers.

DOI: 10.1142/s0217979209053631

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The effect of several eavesdroppers intercept-resend attacks on the quantum error and mutual information between honest parties of a quantum channel chain is investigated within the BB84 (Bennett and Brassard, 1984). The quantum error and the mutual information are computed for arbitrary number of attacks. It is found that the quantum error and the secured–no secured transition depend strongly on the number N of eavesdroppers and their probabilities of intercepting attacks. For N = 3, numerical calculations show that the quantum error exhibits three kinds of different behaviors as a function of the probability of attack ω1of the first eavesdropper namely: (i) the quantum error remains constant for sufficiently small ω1, (ii) exhibits a plateau for intermediate values of ω1, (iii) increases, passes through a maximum and decreases when increasing ω1. However, depending on the probabilities of attack, phase diagrams present several kinds of topologies, in good agreement with the quantum error behavior. Moreover, in the particular case where all eavesdroppers intercept with identical probabilities ω, the quantum error increases as a nonlinear function with the number of eavesdroppers before reaching an upper limit of ≈ 0.475 for sufficiently large N. Besides, it is shown that the secured–no secured transition occurs under the effect of the number of eavesdroppers.

Problem statement: In a previous research, we investigated the quantum key distribution of the well known BB84 protocol with several intercept and resend attacks.In the present research, we studied the effect of many eavesdroppers cloning attacks of the Bennett-Brassard cryptographic protocol on the quantum error and mutual information between honest parties and information with sender for each eavesdropper.Approach: The quantum error and the mutual information were calculated analytically and computed for arbitrary number of cloning attacks.Our objective in this study was to know if the number of the eavesdroppers and their angle of cloning act on the safety of information.Results: It was found that the quantum error and the secured/no secured transition depend strongly on the number of eavesdropper and their angle of attacks.The particular cases where all eavesdroppers collaborate were also investigated.Conclusion: Furthermore, the cloning attack's quantum error is lower than the intercept and resends attacks one, which means that the cloning attacks is the optimal one for arbitrary number of eavesdropper.

The goal of the protocol QKD BB84 is to allow a transmitter and a receiver which uses a quantum channel to exchange their keys and to detect the presence of eavesdropping attacks. In the present research, we investigate the effect of several eavesdroppers with both intercept-resend and cloning attacks. We will propose the different possible cases of the positioning of the eavesdroppers and their strategies of attacks; also we will calculate the mutual information for each case. The explicit expressions of the mutual information and quantum error clearly show that the security of the exchanged information depends on the numbers of the eavesdroppers and their attacks parameters on the quantum channel.

The security analysis of the Ekert 1991 (E91), Bennett 1992 (B92), six-state protocol, Scarani–Acín–Ribordy–Gisin 2004 (SARG04) quantum key distribution (QKD) protocols, and their variants have been studied in the presence of collective-rotation noise channels. However, besides the Bennett–Brassard 1984 (BB84) being the first proposed, extensively studied, and essential protocol, its security proof under collective-rotation noise is still missing. Thus, we aim to close this gap in the literature. Consequently, we investigate how collective-rotation noise channels affect the security of the BB84 protocol. Mainly, we study scenarios where the eavesdropper, Eve, conducts an intercept-resend attack on the transmitted photons sent via a quantum communication channel shared by Alice and Bob. Notably, we distinguish the impact of collective-rotation noise and that of the eavesdropper. To achieve this, we provide rigorous, yet straightforward numerical calculations. First, we derive a model for the collective-rotation noise for the BB84 protocol and parametrize the mutual information shared between Alice and Eve. This is followed by deriving the quantum bit error rate (QBER) for two intercept-resend attack scenarios. In particular, we demonstrate that, for small rotation angles, one can extract a secure secret key under a collective-rotation noise channel when there is no eavesdropping. We observe that noise induced by rotation of 0.35 radians of the prepared quantum state results in a QBER of 11%, which corresponds to the lower bound on the tolerable error rate for the BB84 QKD protocol against general attacks. Moreover, a rotational angle of 0.53 radians yields a 25% QBER, which corresponds to the error rate bound due to the intercept-resend attack. Finally, we conclude that the BB84 protocol is robust against intercept-resend attacks on collective-rotation noise channels when the rotation angle is varied arbitrarily within particular bounds.

We investigate the effect of collective-rotation noise on the security of the six-state quantum key distribution. We study the case where the eavesdropper, Eve, performs an intercept-resend attack on the quantum communication between Alice, the sender, and Bob, the receiver. We first derive the collective-rotation noise model for the six-state protocol and then parameterize the mutual information between Alice and Eve. We then derive quantum bit error rate for three interceptresend attack scenarios. We observe that the six-state protocol is robust against intercept-resend attacks on collective rotation noise channels when the rotation angle is kept within certain bounds.

Abstract: BB84 (Bennett and Brassard 1984) is one of the well known protocols of quantum keys distribution. It is built to allow two interlocutors commonly called Alice and Bob to share two similar binary keys and to detect the eavesdropper presence (the eavesdropper is commonly called Eve). However, Eve presence in a disturbed environment causes errors to the sifted keys and decreases the amount of secure information between Alice and Bob. One of the most important stages in BB84 protocol is to decide by using error probability estimation if we can continue the protocol phases or no. Indeed, this decision is function of some factors like: what is the amount of information that we will lose in the error correction phase? What is the degree of errors detected in our sifted keys? What is the origin of these errors: Eve strategy or the channel disturbance? For these reasons, we will study in this study some conditions to stop BB84 protocol in the context of depolarizing channel. We implement two types of eavesdropping strategy: Intercept and Resend and Cloning Attack.