- Assche, Gilles van. Quantum Cryptography and Secret-Key Distillation. Cambridge University Press, 2006.
- General notes:
- (P65): In order for QKD to be effective, Bob and Alice must be able to communicate over an additional authenticated channel. This channel does not need to be private, but true authentication is required to ensure that Eve cannot perform a man-in-the-middle attack and pretend to be either Bob or Alice while talking to the other. This means that in order for QKD to be truly 100% secure, Alice and Bob must meet once face to face to exchange a key before they can begin communicating over distance.
- Introduction
- Information theory came about in the mid-twentieth century with the intent of answering two fundamental questions:
- What is the fundamental limit of data compression?
- What is the highest possible transmission rate over a communication channel?
- One of the pioneers of information theory, Shannon, was also interested in cryptography. He proved that a perfectly secure cypher would need a secret key that is as long as the message to encrypt. If such a key could be transmitted instantly and securely, there would be no need for further development of cryptography as a perfect solution would have been found. Unfortunately, there is no perfectly confidential way to transmit a key, due to the fact that information is inherently interceptable in some way.
- These days, we have found some ways to send messages encrypted with a small key nearly entirely securely (think trillions of years from every computer in the world to crack), however as proved by Shannon perfect security is impossible unless the secret key is as long as the message.
- A classical computer uses bits, packets of information which can either hold a value of 0 or 1. Quantum computers, on the other hand, use qubits (short for quantum bits) which can be in superposition of 0 and 1 at the same time. This unlocks a greater degree of parallelism without sacrificing performance.
- In 1984, following in the footsteps of Weisner, two scientists proposed a protocol to distribute secret keys using the unique potential of quantum mechanics. They called this protocol quantum key distribution, and it allows the secrecy of a key to be guaranteed by the laws of physics. At a quantum scale, an interception of information will result in a perceptible error in the transmission due to the uncertainty principle, which the two communicating parties will be able to detect. This alone is not enough to guarantee security, because often even if the interception of the key is discovered, it will be too late to prevent the malicious party from using the key to gain access to the encrypted data. This means that some techniques from classical cryptography are still useful to prevent the discovery of the key from being relevant.
- While quantum computers already exist, they are still a ways off from being capable of the incredible feats of computation we know they will one day be able to do. Fortunately, quantum key distribution does not require quantum computing, and is already possible using current technologies such as lasers and fiber optics.
- Quote: “State-of-the-art ciphers, if correctly used, are unbreakable according to today’s knowledge. Unfortunately, their small key size does not offer any long-term guarantee. No one knows what the future will bring, so if clever advances in computer science or mathematics once jeopardize today’s ciphers’ security, quantum key distribution may offer a beautiful alternative solution. Remarkably, the security of quantum key distribution is guaranteed by the laws of quantum mechanics” (3).
- Information theory came about in the mid-twentieth century with the intent of answering two fundamental questions:
- 1.1 A first tour of quantum key distribution
- To perform QKD, any particle which obeys quantum mechanics can be used. The easiest by far, however, is the humble photon, already with innumerable applications in modern science.
- Alice splits up the key into random packets of information which will be sent through the chosen quantum carriers. It is required that the bits she sends are truly random, and that there is no pattern to their distribution.
- While Alice is sending this transmission, Eve might spy on the information. This is not an issue at all, because A) Alice and Bob can detect any interference due to the uncertainty principle and B) Alice and Bob are still able to create a key from the bits which are unknown to Eve.
- 1.1.1 Encoding random bits using qubits
- Any message can be converted to 1s and 0s.
- The state of a qubit is described by 2 complex numbers, the squared absolute values of which always add up to 1.
- 1.1.2 Detecting Eavesdropping
- Upon being measured, a qubit takes on a state based on the measurement result, with no regard to its previous state.
- Whenever Eve measures a photon to discern its value, she has a 1 in 4 chance of changing its state and introducing an error which Bob will detect upon receiving the correct information from Alice.
- 3 – Information Theory
- Definition: “The Shannon entropy (or entropy for short) of a discrete random variable X is denoted by H(X) and is defined as
- 6 – General results on secret-key distillation
- SKD is a way to “distill” a set of shared random variables into a functional and secure secret key. It occurs after key information is sent through QKD, usually over a public channel.
- SKD generally follows two steps.
- First is reconciliation, where the two communicating parties agree on a common string, which may or may not be secret. This is done by Alice sending Bob some redundancy information, denoted in this image as M. It is only during the next step that Eve is removed from the “loop.”
- Next, both parties agree on a hash function and pass their agreed string through it to come up with the secret key K. Unless Eve was able to silently intercept every bit of relevant information during the initial transmission (known to be impossible with true single-particle transfer) then the hashed values generated by Alice and Bob will be completely different from those generated by Eve.
- 10 – The BB84 Protocol
- 10.2 Implementation of BB84
- BB84 is very challenging to implement, regardless of whether a quantum computer is needed. Generating single photons with specific properties alone would have been an insurmountable task just 50 years ago. With today’s technology, however, it is possible to implement BB84.
- It is possible to use a few photons at a time to approximate single photons, with what is called a coherent state. These are very easy to create but follow a Poissonian distribution, meaning that most of the time there will be no photons sent and sometimes more than one. If more than one is sent, it is possible for an eavesdropper to detect this and extract a photon from the pulse without disturbing the stream of information.
- There are two main methods to encode the qubits for BB84:
- A way to encode the qubits is through the polarization of photons. In BB84, only 4 states are required, so vertical, horizontal, and the two superpositions of those (↗ & ↘) can be used, most easily with 4 laser diodes that each emit one type of photon.
- There are many ways for errors in polarization to be introduced when sending photons over a fiber optic channel, so this method of encoding is best suited for open-air communication. It is not impossible, however, and BB84 has successfully been implemented over a 23km fiber (164/128).
- When implementing BB84 over an optical fiber, it is more effective to use phase encoding, which splits a single photon into two “half photons” and measures the points at which they interfere.
- A way to encode the qubits is through the polarization of photons. In BB84, only 4 states are required, so vertical, horizontal, and the two superpositions of those (↗ & ↘) can be used, most easily with 4 laser diodes that each emit one type of photon.
- Photon detectors
- Avalanche photo-diodes are the most commonly-used detectors for measuring the photons coming from Alice. These diodes use a semiconductor such as germanium or silicon with an electrical voltage applied to it. The voltage is high enough that even a single photon hitting the surface causes an “avalanche” of electrons which creates a measurable current.
- 10.2 Implementation of BB84
Quantum Cryptography and Secret-Key Distillation (Book)
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