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arXiv:quant-ph/0107130 · Physical Review Letters

Security of Quantum Key Distribution Using d -Level Systems

Nicolas J. Cerf, Mohamed Bourennane, Anders Karlsson, Nicolas Gisin

Published Mar 8, 2002• 1,355 citations
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Abstract

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.

DOI
10.1103/physrevlett.88.127902

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•Oct 13, 2011•0 citations
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 investigate the security bounds of quantum-cryptographic protocols using ddd-level systems. In particular, we focus on schemes that use two mutually unbiased bases, thus extending the Bennett-Brassard 1984 quantum-key-distribution scheme to higher dimensions. Under the assumption of general coherent attacks, we derive an analytic expression for the ultimate upper security bound of such quantum-cryptography schemes. This bound is well below the predictions of optimal cloning machines. The possibility of extraction of a secret key beyond entanglement distillation is discussed. In the case of qutrits we argue that any eavesdropping strategy is equivalent to a symmetric one. For higher dimensions such an equivalence is generally no longer valid.
We propose an extension of quantum key distribution based on encoding the key into quNits, i.e. quantum states in an N -dimensional Hilbert space. We estimate both the mutual information between the legitimate parties and the eavesdropper, and the error rate, as a function of the dimension of the Hilbert space. We derive the information gained by an eavesdropper using optimal incoherent attacks and an upper bound on the legitimate party error rate that ensures unconditional security when the eavesdropper uses finite coherent eavesdropping attacks. We also consider realistic systems where we assume that the detector dark count probability is not negligible.
We consider a generalization of Ekert's entanglement-based quantum cryptographic protocol where qubits are replaced by NNN- or ddd-dimensional systems (qudits). In order to study its robustness against optimal incoherent attacks, we derive the information gained by a potential eavesdropper during a cloning-based individual attack. In doing so, we generalize Cerf's formalism for cloning machines and establish the form of the most general cloning machine that respects all the symmetries of the problem. We obtain an upper bound on the error rate that guarantees the confidentiality of qudit generalizations of the Ekert's protocol for qubits.
We propose an extension of quantum key distribution based on encoding the key into quNits, i.e. quantum states in an N-dimensional Hilbert space. We estimate both the mutual information between the legitimate parties and the eavesdropper, and the error rate, as a function of the dimension of the Hilbert space. We derive the information gained by an eavesdropper using optimal incoherent attacks and an upper bound on the legitimate party error rate that ensures unconditional security when the eavesdropper uses finite coherent eavesdropping attacks. We also consider realistic systems where we assume that the detector dark count probability is not negligible. PACS numbers: 03.67.Dd, 03.67.Hk, 03.67.−a, 03.65.Ta 1.
In this article, protocols for quantum key distribution based on encoding in higher dimensional systems in N-dimensional Hilbert space are proposed. We extend the original Bennett-Brassard protocol using two complementary bases and two-dimensional states to M mutually complementary bases and N orthogonal vectors in each base. We study the mutual information between the legitimate parties and the eavesdropper and the error rate by considering various incoherent eavesdropping attacks as a function of the dimension of the Hilbert space.
The topic of this thesis is Quantum Cryptography. Based on the laws of Quantum Mechanics, this allows two parties to share a secure key, by using quantum states to carry classical information. Namely, the Quantum Key Distribution (QKD) process, completed with the classical algorithm One Time Pad, leads to an unconditionally secure cryptosystem. Most QKD protocols, like the famous BB84, are realized on unidirectional quantum channels and have probabilistic character. Recently, a new protocol, named LM05, has been proposed, which works on bidirectional quantum channels and in a deterministic way. This thesis is mainly concerned with protocols extending LM05 from binary to d-ary alphabets (dmajor2), by using multi-level quantum systems in d-dimensional Hilbert spaces. The construction of such protocols relies on the notion of Mutually Unbiased Bases (MUB). But the total number of MUB in a Hilbert space of dimension d is known only if d is a prime power. Accordingly, the new protocols are realized under this assumption. As a preliminary step, an explicit expression for MUB encompassing powers of both even and odd primes is discussed. The first proposed extension, called EM09, uses shift operators on MUB to encode information and the usual quantum measurement to realize the control procedure. This guarantees maximal security for dimension d=3 against a powerful individual attack. The second extension, named EM11, is characterized by an innovative control strategy based on a suitable unitary transformation rather than on quantum measurement. Such protocol only works for d an odd prime power, due to the particular choice of the control operator, which is proved to be the only possible with the required properties. The EM11 protocol leads to a relevant improvement. In fact, the security against the same attack is much better than in the EM09 and it increases in terms of the dimension d. Some partial results are also obtained about the possible extensions of the probabilistic protocol called SARG04 to higher dimensions. Finally, it is considered the use of the EM09 and EM11 protocols in the Quantum Direct Communication, where the meaningful message is transmitted without any encryption, as allowed by their deterministic character. In this context, the asymptotic security turns out to be optimal in dimension d=2 for EM09 and in dimension d=3 for EM11.