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arXiv:0802.4160 · Quantum Information and Computation

Deterministic quantum distribution of a d-ary key

Anita Eusebi, Stefano Mancini

Published Nov 1, 2009• 19 citations
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Abstract

We present an extension to a d-ary alphabet of a recently proposed deterministic quantum key distribution protocol. It relies on the use of mutually unbiased bases in prime power dimension d, for which we provide an explicit expression. Then, by considering a powerful individual attack, we show that the security of the protocol is maximal for d=3.

DOI
10.26421/qic9.11-12-3

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We present an extension to a d-ary alphabet of a recently proposed deterministic quantum key distribution protocol. It relies on the use of mutually unbiased bases in prime power dimension d, for which we provide an explicit expression. Then, by considering a powerful individual attack, we show that the security of the protocol is maximal for d=3.
A scalable implementation of (ddd + 1) mutually unbiased bases is proposed for prime power dimensions, where these bases can ensure a security of a robust ddd-dimensional quantum key distribution protocol.
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.
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 introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack tends to one as the numbers c and d increase. In dimension d=2, when the legitimate parties use a complete set of three mutually unbiased bases, the protocol achieves a quantum bit error rate of 57.1%. This represents a significant improvement over the 25% quantum bit error rate achieved in the BB84 protocol or 33% in the six-state protocol.