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European Quantum Electronics Conference (EQEC)

Optimal dimensionality for quantum cryptography

D. B. Horoshko, S. Ya. Kilin

Published Nov 15, 2004• 0 citations
Publisher

Abstract

Two protocols are compared for sharing between Alice and Bob a cryptographic key composed from quarts. The first protocol uses a ququart (4-level system) prepared in one of 5 mutually biased bases. The second one uses two independent qubits, each being prepared in one of 3 mutually unbiased bases. Information gained by the eavesdropper Eve is calculated, suggesting that she uses an asymmetric cloner for cloning individual systems.

DOI
10.1109/eqec.2003.1314275

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Two protocols are compared for sharing between Alice and Bob a cryptographic key composed from quarts. The first protocol uses a ququart (4-level system) prepared in one of 5 mutually biased bases. The second one uses two independent qubits, each being prepared in one of 3 mutually unbiased bases. Information gained by the eavesdropper Eve is calculated, suggesting that she uses an asymmetric cloner for cloning individual systems.
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.
The study of quantum cryptography and quantum entanglement have traditionally been based on two-level quantum systems (qubits). In this paper, we consider a generalization of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by three-level systems (qutrits). In order to investigate the security against the optimal individual attack, we derive the information gained by a potential eavesdropper applying a cloning-based attack. We exhibit the explicit form of this cloner, which is distinct from the previously known cloners, and conclude that the protocol is more robust than those based on entangled qubits as well as unentangled qutrits.
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.
Quantum key distribution based on encoding in three-dimensional systems in the presence of several eavesdroppers is proposed. This extends the BB84 protocol in the presence of many eavesdroppers where two-level quantum systems (qubits) are replaced by three-level systems (qutrits). We discuss the scenarios involving two, three and four complementary bases. We derive the explicit form of Alice and Bob mutual information and the information gained by each eavesdropper. In particular, we show that, in the presence of only one eavesdropper, the protocol involving four bases is safer than the other ones. However, for two eavesdroppers, the security is strongly dependent on the attack probabilities. The effect of a large number of eavesdroppers is also investigated.
The first quantum cryptography protocol, proposed by Bennett and Brassard in 1984 (BB84), has been widely studied in recent years. This protocol uses four states (more precisely, two complementary bases) for the encoding of the classical bit. Recently, it has been noticed that by using the same four states, but a different encoding of information, one can define a protocol which is more robust in practical implementations, specifically when attenuated laser pulses are used instead of single-photon sources [V. Scarani et al., Phys. Rev. Lett. 92, 057901 (2004), referred to as the SARG04 protocol]. We present a detailed study of SARG04 in two different regimes. In the first part, we consider an implementation with a single-photon source: we derive bounds on the error rate QQQ for security against all possible attacks by the eavesdropper. The lower and the upper bound obtained for SARG04 ($Q\ensuremath{\lesssim}10.95%$ and $Q\ensuremath{\gtrsim}14.9%$, respectively) are close to those obtained for BB84 ($Q\ensuremath{\lesssim}12.4%$ and $Q\ensuremath{\gtrsim}14.6%$, respectively). In the second part, we consider a realistic source consisting of an attenuated laser and improve on previous analysis by allowing Alice to optimize the mean number of photons as a function of the distance. The SARG04 protocol is found to perform better than BB84, both in secret-key rate and in maximal achievable distance, for a wide class of Eve's attacks.