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arXiv:1610.07148 · Physical Review A

Revisiting optimal eavesdropping in quantum cryptography: Optimal interaction is unique up to rotation of the underlying basis

Atanu Acharyya, Goutam Paul

Published Feb 21, 2017• 4 citations
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Abstract

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.

DOI
10.1103/physreva.95.022326

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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.
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.
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.
Abstract. The BB84 protocol is used by Alice (the sender) and Bob (the receiver) to settle on a secret classical bit-string by communicating qubits over an insecure quantum channel where Eve (the Eavesdropper) can have access. In this paper, we revisit a well known eavesdropping technique against BB84. We claim that there exist certain gaps in understanding the existing eavesdropping strategyin terms of cryptanalytic view and we try to bridge those gaps in this paper. First we refer to the result where it is shown that in the six-state variant of the BB84 protocol (Bruß, Phys. Rev. Lett., 1998), the mutual information between Alice (the sender) and Eve (the eavesdropper) is higher when two-bit probe is used compared to the one-bit probe and hencethetwo-bitprobeprovidesastrongereavesdroppingstrategy.However, from cryptanalytic point of view, we show that Eve has the same success probability in guessing the bit transmitted by Alice in both the cases of the two-bit and the one-bit probe. Thus, we point out that having higher mutual information may not directly lead to obtaining higher probability in guessing the key bit. It is also explained in the work of Bruß that the six-state variant of the BB84 protocol is more secure than the traditional four-state BB84. We look into this point in more detail and identify that this advantage is only achieved at the expense of communicating more qubits in the sixstate protocol. In fact, we present different scenarios, where given the same number of qubits communicated, the security comparison of the four and six-state protocols is evaluated carefully.
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.
An attempt to eavesdrop on a quantum cryptographic channel reveals itself through errors it inevitably introduces into the transmission. We investigate the relationship between the induced error rate and the maximum amount of information the eavesdropper can extract, in both the two-state B92 [B92 refers to the work of C. H. Bennett, Phys. Rev. Lett. 68, 3121 (1992)] and the four-state BB84 [BB84 refers to the work of C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, New York, 1984), pp. 175--179] quantum cryptographic protocols. In each case, the optimal eavesdropping method that on average yields the most information for a given error rate is explicitly constructed. Analysis is limited to eavesdropping strategies where each bit of the quantum transmission is attacked individually and independently from other bits. Subject to this restriction, however, we believe that all attacks not forbidden by physical laws are covered. Unlike previous work, the eavesdropper's advantage is measured in terms of Renyi (rather than Shannon) information, and with respect only to bits received error-free by Bob (rather than all bits). This alters both the maximum extractable information and the optimal eavesdropping attack. The result can be used directly at the privacy amplification stage of the protocol to accomplish secure communication over a noisy channel.
Quantum cloning can be reviewed as the optimal individual eavesdropping on quantum cryptography. Specifically, the optimal phase-covariant cloning is regarded as the optimal individual attack on the BB84 protocol. In this paper, by exploring Cerf s formalism, we prove the optimality of the asymmetric economical 1→2 phase-covariant quantum cloning in 2 dimensions working without ancilla. In proof, we derive another independent unitary transformation of this cloner. Our result covers previous contributions [Durt T, et al. Phys. Rev. A, 2004, 69:062316].