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Quantum Conference Key Agreement: Difference between revisions
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An ideal CKA protocol, with <math>N</math> users, Alice, Bob<math>_1</math>, Bob<math>_2</math>, ..., Bob<math>_{N-1}</math> should have the following properties: | An ideal CKA protocol, with <math>N</math> users, Alice, Bob<math>_1</math>, Bob<math>_2</math>, ..., Bob<math>_{N-1}</math> should have the following properties: | ||
* '''Correctness:''' A CKA protocol is <math>\epsilon_{corr}</math>-correct if: | * '''Correctness:''' A CKA protocol is said to be ''correct'' if all parties receive the same key at the end of the protocol with high probability. | ||
: Formally, a CKA protocol is <math>\epsilon_{corr}</math>-correct if: | |||
:: <math>p(K_A = K_{B_1} = ... = K_{B_{N-1}}) \geq 1 - \epsilon_{corr}</math> | |||
: where <math>K_A, K_{B_i}</math> are the final keys held by Alice and Bob<math>_i</math>, and <math>p(K_A = K_{B_1} = ... = K_{B_{N-1}})</math> is the probability that all final keys are identical. | : where <math>K_A, K_{B_i}</math> are the final keys held by Alice and Bob<math>_i</math>, and <math>p(K_A = K_{B_1} = ... = K_{B_{N-1}})</math> is the probability that all final keys are identical. | ||
* '''Secrecy:''' A CKA protocol is <math>\epsilon_{sec}</math>-secret if, for <math>\Omega</math> being the event that the protocol does not abort, | * '''Secrecy:''' A CKA protocol is said to be ''secret'' if an eavesdropper Eve cannot differentiate between the key established and a random bitstring. | ||
: Formally, a CKA protocol is <math>\epsilon_{sec}</math>-secret if, for <math>\Omega</math> being the event that the protocol does not abort, | |||
:: <math>p(\Omega)\frac{1}{2}||\rho_{K_AE|\Omega} - \tau_{K_A}\otimes\rho_{E|\Omega}|| \leq \epsilon_{sec}</math>, | |||
: where <math>p(\Omega)</math> is the probability of the event <math>\Omega</math>, <math>\rho_{K_AE|\Omega}</math> is the state shared by Alice and Eve at the end of the protocol given the event <math>\Omega</math>, <math>\tau_{K_A} = \frac{1}{|S|}\sum_{s_i \in S} |s_i\rangle \langle s_i|</math> is the maximally mixed state over all possible values that the key <math>K_A</math> can assume, and <math>S = \{0,1\}^{\times l}</math> where <math>l</math> is the length of the key <math>K_A</math>. | : where <math>p(\Omega)</math> is the probability of the event <math>\Omega</math>, <math>\rho_{K_AE|\Omega}</math> is the state shared by Alice and Eve at the end of the protocol given the event <math>\Omega</math>, <math>\tau_{K_A} = \frac{1}{|S|}\sum_{s_i \in S} |s_i\rangle \langle s_i|</math> is the maximally mixed state over all possible values that the key <math>K_A</math> can assume, and <math>S = \{0,1\}^{\times l}</math> where <math>l</math> is the length of the key <math>K_A</math>. | ||
* '''Completeness:''' A quantum CKA protocol is <math>\epsilon_c</math>-complete if there exists an honest implementation of the protocols, such that the probability of not aborting is greater than <math>1-\epsilon_c</math>. | * '''Completeness:''' A quantum CKA protocol is <math>\epsilon_c</math>-complete if there exists an honest implementation of the protocols, such that the probability of not aborting is greater than <math>1-\epsilon_c</math>. | ||
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* '''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 <math>n_{raw}</math> bits into a secure key of <math> l < n_{raw}</math> bits. | * '''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 <math>n_{raw}</math> bits into a secure key of <math> l < n_{raw}</math> bits. | ||
==Protocols== | |||
* [[Prepare-and-measure Conference Key Agreement]] | |||
* [[Anonymous Conference Key Agreement using GHZ states]] | |||
* [[W-State based Conference Key Agreement]] | |||
* [[Continuous Variable Conference Key Agreement]] | |||
* [[Device Independent Conference Key Agreement]] | |||
<!--==Further Information== --> | <!--==Further Information== --> | ||
<!-- Any issue that could not be addressed or find a place in the above sections or any review paper discussing a feature of various types of protocols related to the functionality. --> | <!-- Any issue that could not be addressed or find a place in the above sections or any review paper discussing a feature of various types of protocols related to the functionality. --> | ||
==References== | ==References== | ||
* [https://arxiv.org/abs/2003.10186 Murta et al.(2020)] discusses the properties of the functionality and provides various example protocols | * [https://arxiv.org/abs/2003.10186 Murta et al.(2020)] discusses the properties of the functionality and provides various example protocols | ||
<div style='text-align: right;'>''*contributed by Chirag Wadhwa''</div> | |||