26 Jul 2001 | Nicolas J. Cerf, Mohamed Bourennane, Anders Karlsson, and Nicolas Gisin
This paper presents two quantum cryptographic protocols using d-level systems (qudits) for secure key distribution. The first protocol uses two mutually unbiased bases, extending the BB84 scheme, while the second uses all d+1 mutually unbiased bases, extending the six-state protocol for qubits. The authors analyze the security of these protocols against both individual and coherent attacks.
For individual attacks, they consider cloning-based strategies, where an eavesdropper (Eve) attempts to clone Alice's qudit and gain information. They derive an upper bound on the error rate that ensures unconditional security against such attacks. They conjecture that the optimal cloner is the best strategy for Eve, making this bound tight. The analysis shows that using two bases results in a slightly lower error rate than using d+1 bases, but the secret key rate is much larger.
For coherent attacks, the authors use an information-theoretic uncertainty principle to derive a lower bound on Bob's information, or an upper bound on the error rate. This provides a sufficient condition for the protocol to generate a nonzero net key rate for all attacks.
The paper also shows that the security of the protocols improves with higher-dimensional systems (larger d), as the disturbance D increases with d. This suggests that higher-dimensional systems are more secure against eavesdropping.
The results are summarized in Table I, which shows the disturbance D = 1 - F (or error rate) as a function of the dimension d. The columns D₂^ind and D_{d+1}^ind display the values of D at which I_AB = I_AE for a cloning-based individual attack with the 2-bases or (d+1)-bases protocol, respectively. The last column D^coh corresponds to an upper bound on D that guarantees security against coherent attacks. The results show that using all d+1 bases provides a slight advantage over using two bases, similar to the six-state protocol for qubits. The paper concludes that the security of quantum cryptography with qudits is more robust for higher-dimensional systems, although practical limitations may affect real-world implementations.This paper presents two quantum cryptographic protocols using d-level systems (qudits) for secure key distribution. The first protocol uses two mutually unbiased bases, extending the BB84 scheme, while the second uses all d+1 mutually unbiased bases, extending the six-state protocol for qubits. The authors analyze the security of these protocols against both individual and coherent attacks.
For individual attacks, they consider cloning-based strategies, where an eavesdropper (Eve) attempts to clone Alice's qudit and gain information. They derive an upper bound on the error rate that ensures unconditional security against such attacks. They conjecture that the optimal cloner is the best strategy for Eve, making this bound tight. The analysis shows that using two bases results in a slightly lower error rate than using d+1 bases, but the secret key rate is much larger.
For coherent attacks, the authors use an information-theoretic uncertainty principle to derive a lower bound on Bob's information, or an upper bound on the error rate. This provides a sufficient condition for the protocol to generate a nonzero net key rate for all attacks.
The paper also shows that the security of the protocols improves with higher-dimensional systems (larger d), as the disturbance D increases with d. This suggests that higher-dimensional systems are more secure against eavesdropping.
The results are summarized in Table I, which shows the disturbance D = 1 - F (or error rate) as a function of the dimension d. The columns D₂^ind and D_{d+1}^ind display the values of D at which I_AB = I_AE for a cloning-based individual attack with the 2-bases or (d+1)-bases protocol, respectively. The last column D^coh corresponds to an upper bound on D that guarantees security against coherent attacks. The results show that using all d+1 bases provides a slight advantage over using two bases, similar to the six-state protocol for qubits. The paper concludes that the security of quantum cryptography with qudits is more robust for higher-dimensional systems, although practical limitations may affect real-world implementations.