by Yee Wei Law - Thursday, 26 October 2023, 2:27 PM
Quantum key distribution (QKD) is a method for generating and distributing symmetric cryptographic keys with information-theoretic security based on quantum information theory [ETS18].
The established key can be however long as required so that it can serve as the key in a one-time pad.
βAuthenticatedβ does not imply βconfidentialβ and does not necessarily require the use of cryptography. Alice and Bob can physically meet and verify each otherβs identity.
In practice, the authenticated classical channel can be established using 1οΈβ£ a pre-shared symmetric key, or 2οΈβ£ public-key cryptography [PAB+20, Sec. F].
A QKD protocol typically proceeds in two phases [Wol21, Ch. 4]:
the classical post-processing phase, where the bitstrings generated in the previous phase are converted into a pair of secret keys.
The security of QKD hinges on the principles of quantum mechanics, rather than the hardness of any computational problem, and hence does not get threatened by advances in computing technologies.
During the quantum transmission phase, any adversary β let us call it π Eve as per tradition β eavesdropping on the quantum channel inherently disturbs the channel and interrupts the key establishment process.
Eve cannot make a copy of any transmitted state (that contributes to the key to be established) thanks to the no-cloning theorem.
It is not possible to perfectly clone an unknown quantum state.
The earliest QKD protocol is due to Bennett and Brassad [BB84] and is called BB84, named after the authors and the year it was proposed.
QKD leverages physical mechanisms, so unavoidably we need to discuss the physical mechanisms that underlie/enable BB84, which are based primarily on the polarisation of photons [Wol21, Sec. 1.3.1].
Polarisation
The polarisation of photons specifies the geometrical orientation of the oscillation of its electromagnetic field.
Polarisation is linear if the field only oscillates in one direction.
Polarisation is circular if the field rotates in a plane as the wave propagates.
We only consider linear polarisation here. For linear polarisation, we distinguish between two bases:
rectilinear basis, which includes horizontal and vertical orientations; and
diagonal basis, which is essentially the rectilinear basis rotated by ; as shown in Fig. 1.
Fig. 1: Polarisation bases and filters [Wol21, Fig. 1.4].
Consider the effect of polarisation filters depicted in Fig. 1:
When a β vertically polarised photon passes through a rectilinear polarisation filter, it is deflected to the right (β‘).
When a β horizontally polarised photon passes through a rectilinear polarisation filter, it is deflected to the left (β¬ ).
When a diagonally polarised photon passes through a rectilinear polarisation filter, it is equally likely to be deflected to the left and right.
Thus, measuring a diagonally polarised photon in the rectilinear basis, and similarly measuring a vertically/horizontally polarised photon in the diagonal basis, give a random result.
We call the rectilinear and diagonal bases mutually conjugate (see Definition 1).
Definition 1: Conjugate bases
Two bases are mutually conjugate [BB84; FGG+97] or unbiased [Wol21, p. 9] if each vector of one basis has equal-length projections onto all vectors of the other basis.
The Heisenberg uncertainty principle ensures that when making two sequential measurements using conjugate bases, the system is disturbed in such a way that the uncertainty of the measurement outcome is maximised [Woj22, p. 42].
With knowledge of polarisation in mind, let us now discuss the quantum transmission phase of BB84, which involves encoding of classical bits into quantum states, communication over a quantum channel, and decoding of quantum states into classical bits.
In the rectilinear basis, 0 and 1 are encoded as β and β respectively.
In the diagonal basis, 0 and 1 are encoded as β and β respectively.
When π§ Bob receives the photons, he randomly (and independently of Alice) decides for each photon whether to measure/decode it in the rectilinear or diagonal basis to retrieve the classical bit.
Since the polarisation state of each photon is a discrete variable, BB84 is an example of a discrete-variable quantum key distribution (DV-QKD) scheme.
The first step is error correction (also called information reconciliation), where they erase all errors in their bit strings. After this step, they hold identical strings.
Direct vs reverse reconciliation [GG02; Djo19, p. 7; PAB+20, Sec. B]
Provides a usable key when the mutual information of Alice () and Bob () exceeds the mutual information of Bob and Eve (), i.e., ; the difference between these two terms gives the asymptotic secret key rate.
The second step is privacy amplification, which is a procedure that minimises Eveβs knowledge of the key.
In terms of performance, a basic figure of merit of every QKD protocol is the secret key rate, i.e. the fraction of secure key bits produced per protocol round.
C. H. Bennett and G. Brassard, Quantum cryptography: Public key distribution and coin tossing, in Proceedings of the International Conference on Computers, Systems & Signal Processing, December 1984, pp. 175β179. Available at https://arxiv.org/abs/2003.06557.
[BBR88]
C. H. Bennett, G. Brassard, and J.-M. Robert, Privacy amplification by public discussion, SIAM Journal on Computing17 no. 2 (1988), 210β229. Available at https://www.proquest.com/docview/919828123.
C. A. Fuchs, N. Gisin, R. B. Griffiths, C.-S. Niu, and A. Peres, Optimal eavesdropping in quantum cryptography. i. information bound and optimal strategy, Phys. Rev. A56 no. 2 (1997), 1163β1172. https://doi.org/10.1103/PhysRevA.56.1163.
[Gra21]
F. Grasselli, Quantum Cryptography: From Key Distribution to Conference Key Agreement, Quantum Science and Technology, Springer Cham, 2021. https://doi.org/10.1007/978-3-030-64360-7.
[GG02]
F. Grosshans and P. Grangier, Reverse reconciliation protocols for quantum cryptography with continuous variables, arXiv preprint quant-ph/0204127, 2002. https://doi.org/10.48550/arXiv.quant-ph/0204127.
S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Advances in Optics and Photonics12 no. 4 (2020), 1012β1236. https://doi.org/10.1364/AOP.361502.