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arXiv:1112.3396 · Physical Review A

Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning

Agnes Ferenczi, Norbert Lütkenhaus

Published May 16, 2012• 93 citations
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Abstract

We investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack where the eavesdropper employs an optimal cloner to attack the quantum key distribution (QKD) protocol. The analysis is done in the context of the security proof in Refs. [Proc. R. Soc. London A 461, 207 (2005); Phys. Rev. Lett. 95, 080501 (2005)] for discrete variable protocols in $d$-dimensional Hilbert spaces. We consider a scenario in which the protocols and cloners are equipped with symmetries. These symmetries are used to define a quantum cloning scenario. We find that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, where we can identify an optimal attack by an optimal cloner. We analyze protocols with 2, $d$ and $d+1$ mutually unbiased bases where $d$ is a prime, and show that for the protocols with 2 and $d+1$ mutually unbiased bases (MUBs) the optimal attack is an optimal cloner but, for the protocols with $d$ MUBs, it is not. Finally, we give criteria to identify protocols which have different signal states, but the same optimal attack. Using these criteria, we present qubit protocols which have the same optimal attack as the Bennett-Brassard 1984 (BB84) protocol or the 6-state protocol.

DOI
10.1103/physreva.85.052310

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We investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack where the eavesdropper employs an optimal cloner to attack the quantum key distribution (QKD) protocol. The analysis is done in the context of the security proof in Refs. [Proc. R. Soc. London A 461, 207 (2005); Phys. Rev. Lett. 95, 080501 (2005)] for discrete variable protocols in ddd-dimensional Hilbert spaces. We consider a scenario in which the protocols and cloners are equipped with symmetries. These symmetries are used to define a quantum cloning scenario. We find that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, where we can identify an optimal attack by an optimal cloner. We analyze protocols with 2, ddd and d+1d+1d+1 mutually unbiased bases where ddd is a prime, and show that for the protocols with 2 and d+1d+1d+1 mutually unbiased bases (MUBs) the optimal attack is an optimal cloner but, for the protocols with ddd MUBs, it is not. Finally, we give criteria to identify protocols which have different signal states, but the same optimal attack. Using these criteria, we present qubit protocols which have the same optimal attack as the Bennett-Brassard 1984 (BB84) protocol or the 6-state protocol.
In this thesis we develop practical tools for quantum key distribution (QKD) security proofs. We apply the tools to provide security proofs for several protocols, ranging from discrete variable protocols in high dimensions, protocols with realistic implementations, measurement device independent QKD and continuous-variable QKD. The security proofs are based on the Devetak-Winter security framework [I. Devetak and A. Winter, Proc. of the Roy. Soc. of London Series A, 461, 207 (2005); B. Kraus, N. Gisin, and R. Renner, Phys. Rev. Lett., 95 080501 (2005)]. \n \nIn the key rate calculation, it is often convenient to assume that the optimal attack is symmetric. \nUnder the assumption that the parameter estimation is based on coarse-grained observations, we show that the optimal attack is symmetric, if the protocol and the postselection have sufficient symmetries. \nAs an example we calculate the key rates of protocols using 2, d and d+1 mutually unbiased bases in d-dimensional Hilbert spaces. \n \nWe investigate the connection between the optimal collective eavesdropping attack and the optimal cloning attack, in which the eavesdropper employs an optimal cloner to attack the protocol. \nWe find that, in general, it does not hold that the optimal attack is an optimal cloner. However, there are classes of protocols, for which we can identify the optimal attack by an optimal cloner. \nWe analyze protocols with mutually unbiased bases in d dimensions, and show that for the protocols with 2 and d+1 mutually unbiased bases the optimal attack is an optimal cloner, but for the protocols with d mutually unbiased bases, it is not. \n \nIn optical implementations of the phase-encoded BB84 protocol, the bit information is usually encoded in the phase of two consecutive photon pulses generated in a Mach-Zehnder interferometer. In the actual experimental realization, the loss in the arms of the Mach-Zehnder interferometer is not balanced, for example because only one arm contains a lossy phase modulator. \nSince the imbalance changes the structure of the signals states and measurements, the BB84 security analysis no longer applies in this scenario. \nWe provide a security proof for the unbalanced phase-encoded BB84. The loss does lower the key rate compared to a protocol without loss. However, for a realistic parameter regime, the same key rate is found by applying the original BB84 security analysis. \n \nRecently, the security of a measurement device-independent QKD setup with BB84 signal states was proven in Refs. [H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett., 108 (2012); S. L. Braunstein and S. Pirandola, Phys. Rev. Lett., 108 (2012)]. In this setup Alice and Bob send quantum states to an intermediate node, which performs the measurement, and is assumed to be controlled by Eve. \nWe analyze the security of a measurement device-independent QKD protocol with B92 signal states, and calculate the key rates numerically for a realistic implementation. \nBased on our security proof we were able to prove the security of the strong reference pulse B92 protocol. \n \nWe analyze the security of continuous-variable protocols using the entropic uncertainty relations established in Ref. [M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, Nature Physics 6, 659 (2010)] to provide an estimate of the key rate based on the observed first and second moments. We analyze a protocol with squeezed coherent states and the 2-state protocol with two coherent states with opposite phases.
The security of quantum cryptography is guaranteed by the no-cloning theorem, which implies that an eavesdropper copying transmitted qubits in unknown states causes their disturbance. Nevertheless, in real cryptographic systems some level of disturbance has to be allowed to cover, e.g., transmission losses. An eavesdropper can attack such systems by replacing a noisy channel by a better one and by performing approximate cloning of transmitted qubits which disturb them but below the noise level assumed by legitimate users. We experimentally demonstrate such symmetric individual eavesdropping on the quantum key distribution protocols of Bennett and Brassard (BB84) and the trine-state spherical code of Renes (R04) with two-level probes prepared using a recently developed photonic multifunctional quantum cloner [Lemr et al., Phys. Rev. A 85, 050307(R) (2012)]. We demonstrated that our optimal cloning device with high-success rate makes the eavesdropping possible by hiding it in usual transmission losses. We believe that this experiment can stimulate the quest for other operational applications of quantum cloning.
We consider two quantum cryptographic schemes relying on encoding the key into qudits, i.e., quantum states in a d-dimensional Hilbert space. The first cryptosystem uses two mutually unbiased bases (thereby extending the BB84 scheme), while the second exploits all d+1 available such bases (extending the six-state protocol for qubits). We derive the information gained by a potential eavesdropper applying a cloning-based individual attack, along with an upper bound on the error rate that ensures unconditional security against coherent attacks.
We study a general quantum key distribution protocol in higher dimension. In this protocol, quantum states in arbitrary g+1g+1g+1 ($1\ensuremath{\le}g\ensuremath{\le}d$) out of all d+1d+1d+1 mutually unbiased bases in a ddd-dimensional system can be used for the key encoding. This provides a natural generalization of the quantum key distribution in higher dimension and recovers the previously known results for g=1g=1g=1 and ddd. In our investigation, we study Eve's attack by two slightly different approaches. One is considering the optimal cloner of Eve, and the other, defined as the optimal attack, is maximizing Eve's information. We derive results for both approaches and show the deviation of the optimal cloner from the optimal attack. With our systematic investigation of the quantum key distribution protocols in higher dimension, one may balance the security gain and the implementation cost by changing the number of bases in the key encoding. As a side product, we also prove the equivalency between the optimal phase covariant quantum cloning machine and the optimal cloner for the $g=d\ensuremath{-}1$ quantum key distribution.
Many modern classical encryption schemes, based on computational complexity, are threatened by quantum computers. Quantum Key Distribution (QKD) aims to address this threat, with security enforced by quantum-mechanical principles. The security of QKD protocols, like the six-state protocol (6QKD), relies on understanding all possible eavesdropping attacks. The optimal known individual eavesdropping attack on 6QKD in a noiseless channel is the Universal Quantum Cloning Machine (UQCM). However, the utility of quantum machine learning (QML) for QKD eavesdropping analysis is largely unexplored. The recent work by Decker et al. (2025) used QML to find new attacks on the four-state BB84 protocol, but this methodology has not yet been generalized to the more complex 6QKD protocol. This thesis generalizes the quantum circuit (QCL) learning framework to analyze individual eavesdropping attacks on 6QKD. We designed a parameterized quantum circuit (ansatz) appropriate for 6QKD, notably adapting our circuit to two ancilla, and expanding the measurement strategy to three bases. We used QCL to find the optimal attack parameters in a simulated noiseless, asymmetric (bit-flip, phase-flip), and symmetric (depolarizing) noise channels. In the noiseless quantum channel, the QCL successfully converges to and reproduces the theoretical performance of the UQCM, validating the adapted methodology. In the asymmetric bit-flip and phase-flip noise channels, the QCL discovers a novel "imbalanced cloner" attack that outperforms the standard UQCM in average fidelity. The model learns to exploit the noise asymmetry by sacrificing the fidelity in the degraded bases and gaining higher fidelity in the unaffected basis. In a symmetric depolarizing noise channel, the QCL performance converges exactly to the UQCM, demonstrating it finds no exploitable noise asymmetry. This work provides the first QCL-based security analysis of the 6QKD protocol. We demonstrate that the QCL is a generalizable tool for quantum security analysis. Furthermore, we present the first (to our knowledge) evidence of a 6QKD "imbalanced cloner" that outperforms UQCM in asymmetric noisy environments.
We analyze the security of an Ekert type bipartite key distribution protocol coming under cloning attacks by an untrusted observer. Precisely, an eavesdropper is assumed to intercept and apply an asymmetric, phase-variant cloner on one or both qubits which are transmitted over public channel from an authenticated source to the trusted parties. We partially characterize two-qubit states from the perspective of designing a QKD protocol that can generate a distillable secret key among the trusted users even under such cloning attacks. We rely upon Devetak Winter framework for computing key rate so as to identify states, which when used in the protocol, secret-key can be distilled. Utility of an entangled state, for designing a successful QKD protocol, changes from single-qubit to double-qubits cloning attacks. Interestingly, we can distinguish some useful families of two-qubit entangled states when only one of the transmitted qubits is cloned by a malicious third party.