Relativistic quantum cryptography is a sub-field of quantum cryptography, in which in addition to exploiting the principles of quantum physics, the no-superluminal signalling principle of relativity theory stating that information cannot travel faster than light is exploited too. Technically speaking, relativistic quantum cryptography is a sub-field of relativistic cryptography, in which cryptographic protocols exploit the no-superluminal signalling principle, independently of whether quantum properties are used or not. However, in practice, the term relativistic quantum cryptography is used for relativistic cryptography too.
History In 1997 and 1998, some important tasks in mistrustful cryptography were shown to be impossible to achieve with unconditional security. Mayers and Lo and Chau showed that unconditionally secure quantum bit commitment was impossible. Lo showed that oblivious transfer and a broad class of secure computations were also impossible to achieve with unconditional security in quantum cryptography. Moreover, Lo and Chau showed that unconditionally secure ideal quantum coin tossing was impossible too. In this context, Kent provided in 1999 the first relativistic cryptographic protocols, for bit commitment and ideal coin tossing, which overcome the assumptions made by Mayers, Lo and Chau, and achieve unconditional security. Since then, other unconditionally secure relativistic protocols for bit commitment have been found by Kent and others, and other cryptographic tasks have been investigated in the setting of relativistic quantum cryptography.
Basics
No-signalling and no-superluminal signalling The no-signalling principle of quantum theory states that information cannot be communicated between two distinct locations L0 and L1 without the transmission of any physical systems, despite any quantum entanglement shared between L0 and L1. This implies, in particular, that without the transmission of any physical systems between L0 and L1, quantum correlation between L0 and L1 cannot be used to transmit information between L0 and L1, even if they are non-locally causal and violate Bell inequalities. According to relativity theory, physical systems cannot travel faster than the speed of light. Thus, it follows from the no-signalling principle that information cannot travel faster than the speed of light. This is called the no-superluminal signalling principle. The principle of no-superluminal signalling is the key physical principle exploited in relativistic cryptography. It guarantees that the outcome x of a random variable X obtained at some spacetime point P cannot influence the probability that a random variable Y takes some value y at a spacelike separated spacetime point Q. Thus, for example, if two parties Alice and Bob have each two agents, with the first agent of Bob sending a secret message x to a first agent of Alice at the spacetime point P, and with the second agent of Alice sending a secret message y to the second agent of Bob at the spacetime point Q, with P and Q spacelike separated, then Bob can be guaranteed that the message y received from Alice was chosen independently of the message x that he gave Alice, and vice versa. This is a useful mathematical property that is exploited to prove the security of cryptographic protocols in relativistic cryptography.
The setting It is a fundamental requirement in relativistic cryptography that the parties implementing the cryptographic task have a good description of spacetime, at least within the region of spacetime where the task is implemented. For example, in protocols implemented near the Earth surface, it can be assumed that spacetime is close to Minkowski. Importantly, this means that, near the Earth surface, physical systems and information cannot travel faster than the speed of light through vacuum, which is approximately 300,000 km/s. In principle, relativistic cryptography can be applied with more general spacetimes, as long as the parties can guarantee that there are no mechanisms allowing instant communication, like wormholes. Another requirement is that the parties have access to a common reference frame, so that they can guarantee that some communication events are spacelike separated. In relativistic cryptography, it is assumed that each party participating in the cryptographic task has various trusted agents that collaborate to implement the task. The agents implement the protocol by performing different actions at various points in spacetime. The agents of the same party may communicate via authenticated and secure channels, which can be implemented with previously shared secure keys, for example using one-time pads. Various tasks investigated by relativistic cryptography consist in tasks of mistrustful cryptography, in which two or more mistrustful parties must collaborate to implement a cryptographic task while at the same time being guaranteed that other parties do not cheat. Examples of tasks in mistrustful cryptography are bit commitment, coin tossing, oblivious transfer and secure computations. Key distribution does not belong to mistrustful cryptography, because in this case the parties distributing the key trust each other. In relativistic cryptography, each participating party has various trusted agents, who collaborate with each other by performing different actions at various spacetime points. For example, Alice and Bob can be two companies with offices and laboratories at various locations in the Earth. Alice's offices and laboratories work in collaboration and trust each other. Similarly, Bob's offices and laboratories work in collaboration and trust each other. But Alice and Bob do not trust each other.
Tasks investigated in relativistic cryptography
Bit commitment
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