Quantum Conference Key Agreement

Functionality

Conference key agreement (CKA), or multipartite key distribution is a cryptographic task where more than two parties wish to establish a common secret key. It is possible to compose bipartite QKD protocols to accomplish this task. However, protocols based on multipartite quantum correlations may be more efficient and practical in future large scale quantum networks.

Tags: Multi Party Protocols, Quantum Enhanced Classical Functionality, Specific Task

Properties

An ideal CKA protocol, with $N$ users, Alice, Bob$_{1}$, Bob$_{2}$, ..., BobFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{N-1}} should have the following properties:

• Correctness: A CKA protocol is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{corr}} -correct if:

  Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(K_A = K_{B_1} = ... = K_{B_{N-1}}) \geq 1 - \epsilon_{corr}}

where $K_{A},K_{B_{i}}$ are the final keys held by Alice and Bob$_{i}$, and $p(K_{A}=K_{B_{1}}=...=K_{B_{N-1}})$ is the probability that all final keys are identical.

• Secrecy: A CKA protocol is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_{sec}} -secret if, for $\Omega$ being the event that the protocol does not abort,

  $p(\Omega ){\frac {1}{2}}||\rho _{K_{A}E|\Omega }-\tau _{K_{A}}\otimes \rho _{E|\Omega }||\leq \epsilon _{sec}$,

where $p(\Omega )$ is the probability of the event $\Omega$, $\rho _{K_{A}E|\Omega }$ is the state shared by Alice and Eve at the end of the protocol given the event $\Omega$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau_{K_A} = \frac{1}{|S|}\sum_{s_i \in S} |s_i\rangle \langle s_i|} is the maximally mixed state over all possible values that the key $K_{A}$ can assume, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \{0,1\}^{\times l}} where $l$ is the length of the key $K_{A}$.

• Completeness: A quantum CKA protocol is $\epsilon _{c}$-complete if there exists an honest implementation of the protocols, such that the probability of not aborting is greater than $1-\epsilon _{c}$.

Generic Protocol Structure

• Preparation and distribution: A source distributes a multipartite entangled state to the $N$ parties. This step is repeated $n$
• Measurements: Upon receiving the systems, the parties perform local measurements and record the classical outcome. The measurements are randomly chosen according to the specifications of the protocol. One of the possible measurement settings is used with higher probability and is called the key generation measurements. The other measurements are used for test rounds, which only occasionally occur.
• Parameter estimation: The parties announce the inputs and outputs of their test rounds and of some randomly chosen key generation rounds which are used to estimate their correlation and the potential influence of an eavesdropper. At the end of this step, each party is left with a string of $n_{raw}<n$ bits, which constitute their raw key.
• Information reconciliation (error correction): The parties publicly exchange classical information in order for the Bobs to correct their raw keys to match Alice's string. In the multipartite case, the information reconciliation protocol needs to account for the correction of the strings of all the Bobs.
• Privacy amplification: Alice randomly picks a hash function, chosen among a two-universal family of hash functions, and communicates it to the Bobs. Every party applies the hash function to turn her/his partially secure string of $n_{raw}$ bits into a secure key of $l<n_{raw}$ bits.

References

• Murta et al.(2020) discusses the properties of the functionality and provides various example protocols