Quantum coin flipping

Quantum coin flipping uses the principles of quantum mechanics to encrypt messages for secure communication. Unlike in other types of quantum cryptography, quantum coin flipping is a protocol used between two users who do not trust each other.[1] Since the players do not trust each other, they both want to win the coin toss and, thus, they will try to cheat in various ways.[1]

Quantum coin flipping and other types of quantum cryptography communicate information through the transmission of qubits. The accepting player does not know the information in the qubit until he performs a measurement.[2] The security of quantum cryptography is in the qubits. In each of the qubits, information is carried by a single photon.[3] Once the receiving player measures the photon, it is altered and will not measure the same way again.[3] The fact that a photon can only be read the same way once allows for an easy detection of any other party attempting to intercept the message.[3]

Quantum coin flipping is a secure means of communicating in theory, but difficult to accomplish.[1][3]

History

Manuel Blum introduced coin flipping as part of a classical system in 1983.[4] Blum’s coin flipping protocol was based on computational algorithms and assumptions.[4] Blum’s version of coin flipping answers the cryptographic problem in the example:

Alice and Bob are recently divorced, living in two separate cities, and want to decide who gets to keep the car. To decide, Alice wants to flip a coin over the telephone. However, Bob is concerned that if he were to tell Alice heads, she would flip the coin and automatically tell him that he lost.[2]

Thus, the problems with Alice and Bob are that they do not trust each other, the only resource they have is the telephone communication channel, and there is not a third party available to read the coin.[2] The only way that Alice and Bob can exchange information is if they are both being truthful and agree on a value or until one of them is convinced that the other is cheating.[2]

In 1984, quantum cryptography emerged in the paper written by Charles H. Bennett and Giles Brassard. In this paper, the two introduced the idea of using quantum mechanics to enhance previous cryptographic protocols such as coin flipping.[1] Since 1984, many researchers have applied quantum mechanics to cryptography as they have proven theoretically to be more secure than classical cryptography.[1] However, demonstrating these protocols, like quantum coin flipping, in practical systems has been difficult to accomplish.[1]

As published in 2014, a group of scientists at the Laboratory for Communication and Processing of Information (LTCI) in Paris have implemented quantum coin flipping protocols experimentally.[1] The researchers have reported that the protocol performs better than a classical system over a suitable distance for a metropolitan area optical network.[1]

Coin flipping protocol

Quantum coin flipping is when random qubits are generated between two players that do not trust each other because both of them want to win the coin toss, which could lead them to cheat in a variety of ways.[1] The essence of coin flipping occurs when the two players issue a sequence of instructions over a communication channel that then eventually results in an output.[3]

A basic quantum coin flipping protocol involves two people: Alice and Bob.[5]

  1. Alice sends Bob a set number of Κ photon pulses in the quantum states | Φαici>. Each of these photon pulses is independently prepared following a random choice by Alice of basis αi and bit ci where i = 1, 2, 3…Κ.
  2. Bob then measures the pulses from Alice by identifying a random basis βi. Bob records these photons and then reports back the first successfully measured photon j to Alice along with a random bit b.
  3. Alice reveals the basis and bit that she used at the basis Bob gave her. If the two bases and bits match, then both parties are truthful and can exchange information. If the bit reported by Bob is different than that of Alice’s, one is not being truthful.
Alice decides her random basis and sequence of qubits. She then sends the qubits as photons to Bob via the quantum channel. Bob detects these qubits and records his results in a table. Based on the table, Bob makes his guess to Alice on what basis she used.

A more general explanation of the above protocol is as follows:[6]

  1. Alice first chooses a random basis (such as diagonally) and a sequence of random qubits. Alice then encodes her chosen qubits as a sequence of photons following the chosen basis. She then sends these qubits as a train of polarized photons to Bob through the communication channel.
  2. Bob chooses a sequence of reading bases randomly for each individual photon. He then reads the photons and records the results in two tables. One table is of the rectilinear (horizontal or vertical) received photons and one of the diagonally received photons. Bob may have holes in his tables due to losses in his detectors or in the transmission channels. Based on this table, Bob makes a guess as to which basis Alice used and announces his guess to Alice. If he guessed correctly, he wins and if not, he loses.
  3. Alice reports whether he won or not by announcing what basis she used to Bob. Alice then confirms the information by sending Bob her entire original qubit sequence that she used in step 1.
  4. Bob compares Alice’s sequence with his tables to confirm that no cheating occurred on Alice’s part. The tables should correspond to Alice’s basis and there should be no correlation with the other table.

Cheating

The key issue with coin flipping is that it occurs between two distrustful parties.[6] These two parties are communicating through the communication channel some distance from each other and they have to agree on a winner or loser with each having a 50 percent chance of winning.[6] However, since they are distrustful of one another, cheating is likely to occur. Cheating can occur in a number of ways such as claiming they lost some of the message when they do not like the result or increasing the average number of photons contained in each of the pulses.[1]

For Bob to cheat, he would have to be able to guess Alice’s basis with a probability greater than ½.[6] In order to accomplish this, Bob would have to be able to determine a train of photons randomly polarized in one basis from a train of photons polarized in another basis.[6]

Alice, on the other hand, could cheat in a couple of different ways, but she has to be careful because Bob could easily detect them as cheating.[6] When Bob sends a correct guess to Alice, she could convince Bob that her photons are actually polarized the opposite of Bob’s correct guess.[6] Alice could also send Bob a different original sequence than she actually used in order to beat Bob.[6]

Detecting a third-party

Single photons are used to pass the information from one player to the other (qubits).[3] In this protocol, the information is encoded in the single photons with polarization directions of 0, 45, 90, and 135 degrees, non-orthogonal quantum states.[6] When a third party attempts to read or gain information on the transmission, they alter the photon’s polarization in a random way that is likely detected by the two players because it does not match the pattern exchanged between the two legitimate users.[6]

Implementation

Experimental

As mentioned in the history section, scientists at the LTCI in Paris have experimentally carried out a quantum coin flipping protocol, giving an example of using quantum coin flipping. Previous protocols called for a single photon source or an entangled source to be secure. However, these sources are why it is difficult for quantum coin flipping to be implemented. Instead the researchers at LTCI used the effects of quantum superposition rather than a single photon source, which they claim makes implementation easier with the standard photon sources available.[1]

These researchers used the Clavis2 platform developed by IdQuantique for their protocol. The researchers had to modify the Clavis2 system to get it to work for the coin flipping protocol. The experimental setup they used with the Clavis2 system, involves a two-way approach. Light pulsed at 1550 nanometres is sent from Bob to Alice. Alice then uses a phase modulator to encrypt the information. After encryption, she then uses a Faraday mirror to reflect and attenuate the pulses at her chosen level and sends them back to Bob. Using two high quality single photon detectors, Bob chooses a measurement basis in his phase modulator to detect the pulses from Alice.[5]

They replaced the detectors on Bob’s side because of the low detection efficiencies of the previous detectors. When they replaced the detectors, they were able to show a quantum advantage on a channel for over 15 kilometres (9.3 mi). A couple of other challenges the group faced was reprogramming the system because photon source attenuation was high and performing system analyses to identify losses and errors in system components. With these corrections, the scientists were capable of implementing a coin flipping protocol by introducing a small honest abort probability, the probability that two honest participants cannot obtain a coin flip at the end of the protocol, but at a short communication distance.[1]

References

  1. 1 2 3 4 5 6 7 8 9 10 11 12 Stuart Mason Dambort, "Heads or tails: Experimental quantum coin flipping cryptography performs better than classical protocols", Phys.org, March 26, 2014
  2. 1 2 3 4 C. Döscher and M. Keyl, "An Introduction to Quantum Coin-Tossing", Cornell University Library, February 1, 2008
  3. 1 2 Andris Ambainis, "A New Protocol and Lower Bounds for Quantum Coin Flipping", Cornell University Library, April 4, 2002
  4. 1 2 Anna Pappa et al., "Experimental Plug and Play Quantum Coin Flipping", Nature Communications, April 24, 2014
  5. 1 2 3 4 5 6 7 8 9 10 Charles H. Bennett and Giles Brassard, "Quantum cryptography: Public key distribution and coin tossing", Theoretical Computer Science, December 4, 2014
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