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Quantum Cryptography

  • Reference work entry
Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

Quantum Key Distribution: Motivation and Introduction

Security Proofs

Experimental Fundamentals

Experimental Implementation of BB84 Protocol

Other Quantum Key Distribution Protocols

Quantum Hacking

Beyond Quantum Key Distribution

Future Directions

Acknowledgments

Bibliography

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Abbreviations

One-time pad:

One‐time pad is a classical encryption algorithm invented by Gilbert Vernam in 1917. In one‐time pad algorithm, the legitimate users share a random key (e. g. a random binary string) that is not known to anyone else. The message is combined with this random key (“pad”) which is as long as the message. The key is used only once (“one‐time”). The most typical usage is in binary case, where an XOR operation is applied between the message and the key to achieve the ciphertext.Claudé Shannon proved that the one‐time pad provides perfect secrecy in 1949. The perfect secrecy is defined that the ciphertext does not give any additional information on the message.

Key distribution problem:

The key distribution problem originates from the one‐time pad encryption. The one‐time pad encryption requires that the two parties share a secret random key before the communication. This key is usually generated by one party. The key distribution problem is how to distribute this random key from one party to the other party securely. This problem is non‐solvable classically, but is solvable via quantum key distribution.

Quantum key distribution:

Quantum key distribution (QKD) is a method to distribute a random key between two parties securely. The main idea is to encode the bit value on the quantum state of certain particle (usually photon) and send the particle to the receiver. The quantum no‐cloning theorem guaranteed that any eavesdropper cannot duplicate the encoded quantum information perfectly.

BB84:

BB84 is the first and so far the most popular quantum cryptography protocol. It was proposed by C. H. Bennet and G. Brassard in 1984 [1]. In the original BB84 proposal, the quantum information is encoded on the polarizations of photons. Later BB84 was extended to the phase coding. Detailed description of BB84 protocol can be found in Sect. “Introduction”.

B92:

B92 is a quantum cryptography protocol proposed by C. H. Bennet in 1992 [2]. It uses two non‐orthogonal states (e. g. the horizontally – and 45\( { ^\circ } \) polarized photons) to denote “0” and “1”. It is simpler than BB84 protocol in implementation.

E91:

E91 is a quantum cryptography protocol proposed by A. Ekert in 1991 [3]. It is based on entangled photon pairs. E91 protocol is often used in free‐space quantum key distribution.

Uni‐directional QKD:

Uni‐directional QKD is the QKD scheme in which Alice (sender) generates the photon, encodes the quantum information on it, and sends it to Bob (receiver).

Bi‐directional QKD, or “Plug & play” QKD:

Bi‐directional QKD, or “Plug & play” QKD is the QKD scheme in which Bob generates strong laser pulses and sends them to Alice.Alice encodes her quantum information on the pulse and attenuates the pulse to single‐photon level, and sends it back to Bob through the same channel. This design can automatically compensate the phase and the polarization drifting in the channel.

Single photon source:

Single photon source is the light source that can generate a single photon on demand. A perfect single photon source should have zero probability to generate multi photons once triggered. Single photon source is required in the original BB84 protocol. However, it is no longer under absolute demand due to the discovery and implementation of decoy state QKD.

Fainted laser source:

Fainted laser source is the light source that has a standard pulsed laser source and a heavy attenuator. The average output photon number is usually set to \( { \sim0.1 } \) photon per pulse. This low average photon number is to suppress the production of multi photon signals. However, due to the poisson nature of laser source, the probability of multi photon production can never reach zero unless the laser is turned off.

Single photon detector:

Single photon detector is sensitive to the weakest light signals – signals with single photons. Most single photon detectors are threshold by means that they can only detect the arrival of one or more photons, but cannot count the number of photons within one signal.

Dark count:

Dark count is the event that the detector reports a detection while no photon actually hits it. It is a key parameter for single photon detectors.Dark count becomes important when the channel loss between the sender and the receiver is high (i. e., when very few photons can reach the receiver).

Qubit:

Qubit (or quantum bit) is the fundamental unit of quantum information. Whereas a classical bit can take value of either “0” or “1”, a qubit can take a value in any superposition of two distinguishable (i. e., orthogonal) states commonly labeled by \( { | 0 \rangle } \) and \( { | 1 \rangle } \).In other words, a (pure) qubit state can be written in the form \( { a | 0 \rangle + b | 1 \rangle } \) where a and b are complex numbers. The normalization constraint is that \( { |a|^2 + |b|^2 = 1 } \).Physically, a qubit can be encoded in any two‐level quantum system, such as the two polarization of a single photon or two atomic levels of an atom.

Bit-flip:

Bit‐flip is a typical noise in both classical and quantum communication. In quantum cryptography, a bit‐flip in the channel will transform an initial state \( | i \rangle=a| 0 \rangle+ b| 1 \rangle \) into the final state \( | f \rangle=a| 1 \rangle+b| 0 \rangle \).

Phase‐flip:

Phase‐flip is a typical noise that is unique in quantum communication. A phase‐flip in the quantum channel will transform an initial state \( | i \rangle=a| 0 \rangle+ b| 1 \rangle \) into the final state \( | f \rangle=a| 0 \rangle-b| 1 \rangle \).

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Acknowledgments

We thank various funding agencies including NSERC, CRC program, QuantumWorks, CIFAR, MITACS, CIPI, PREA, CFI, and OIT for their financialsupport.

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  1. Center for Quantum Information and Quantum Control, Department of Physics and Department of Electrical and Computer Engineering, University of Toronto, Toronto, Canada

    Hoi-Kwong Lo & Yi Zhao

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Lo, HK., Zhao, Y. (2012). Quantum Cryptography. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_151

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