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Optimal Eavesdropping in Quantum Cryptography: Choice of Interaction Is Unique up to a Rotation of the Underlying Basis

Atanu Acharyya

Published Sep 1, 2016• 0 citations
OpenAlex

Abstract

A general framework of optimal eavesdropping on BB84 protocol was provided by Fuchs et al. in 1997. An upper bound on mutual information was derived in their work, which could be achieved by a specific type of interaction and the corresponding measurement. However, uniqueness of an optimal interaction was left as an open problem there. We resolve this problem here and establish the uniqueness. Nevertheless, the description of an optimal interaction changes as the basis of description gets rotated. The specific choice of optimal interaction by Fuchs et al. is shown to be a special case of the form provided in our work.

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Attacks of BB84 protocol in quantum cryptography

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A general framework of optimal eavesdropping on BB84 protocol was provided by Fuchs et al. in 1997. An upper bound on mutual information was derived in their work, which could be achieved by a specific type of interaction and the corresponding measurement. However, uniqueness of an optimal interaction was left as an open problem there. We resolve this problem here and establish the uniqueness. Nevertheless, the description of an optimal interaction changes as the basis of description gets rotated. The specific choice of optimal interaction by Fuchs et al. is shown to be a special case of the form provided in our work.
A general framework of optimal eavesdropping on BB84 protocol was provided by Fuchs et al. [Phys. Rev. A 56, 1163 (1997)]. An upper bound on mutual information was derived, which could be achieved by a specific type of interaction and the corresponding measurement. However, uniqueness of optimal interaction was posed as an unsolved problem there and it has remained open for almost two decades now. In this paper, we solve this open problem and establish the uniqueness of optimal interaction up to rotation. The specific choice of optimal interaction by Fuchs et al. is shown to be a special case of the form derived in our work.
We consider the Bennett-Brassard cryptographic scheme, which uses two conjugate quantum bases. An eavesdropper who attempts to obtain information on qubits sent in one of the bases causes a disturbance to qubits sent in the other basis. We derive an upper bound to the accessible information in one basis, for a given error rate in the conjugate basis. Independently fixing the error rates in the conjugate bases, we show that both bounds can be attained simultaneously by an optimal eavesdropping probe. The probe interaction and its subsequent measurement are described explicitly. These results are combined to give an expression for the optimal information an eavesdropper can obtain for a given average disturbance when her interaction and measurements are performed signal by signal. Finally, the relation between quantum cryptography and violations of Bell's inequalities is discussed.
We consider the Bennett-Brassard cryptographic scheme, which uses two conjugate quantum bases. An eavesdropper who attempts to obtain information on qubits sent in one of the bases causes a disturbance to qubits sent in the other basis. We derive an upper bound to the accessible information in one basis, for a given error rate in the conjugate basis. Independently fixing the error rate in the conjugate bases, we show that both bounds can be attained simultaneously by an optimal eavesdropping probe, consisting of two qubits. The qubits' interaction and their subsequent measurement are described explicitly. These results are combined to give an expression for the optimal information an eavesdropper can obtain for a given average disturbance when her interaction and measurements are performed signal by signal. Finally, the relation between quantum cryptography and violations of Bell's inequalities is discussed.
We examine a situation that nnn eavesdroppers attack the Bennett-Brassard cryptographic protocol via their own optimal and symmetric strategies. Information gain and mutual information with sender for each eavesdropper are explicitly derived. The receiver's error rate for the case of arbitrary nnn eavesdroppers can be derived using a recursive relation. Although the first eavesdropper can get mutual information without disturbance arising due to other eavesdroppers, subsequent eavesdropping generally increases the receiver's error rate. Other eavesdroppers cannot gain information on the input signal sufficiently. As a result, the information each eavesdropper gains becomes less than optimal one.