Construction of a shared secret key using continuous variables

Abstract
Motivated by recent advances in quan- tum cryptography with continuous variables, we study the problem of extracting a shared digital secret key from two correlated real values. Alice has access to a real value , and Bob to another value such that . They wish to convert their values into a shared secret digi- tal information while leaking as little information as possi- ble to Eve. We show how the problem can be decomposed in two subproblems known in other contexts. The first is the design of a quantizer that maximizes a mutual infor- mation criterion, the second is known as coding with side information. I. INTRODUCTION The work presented in this paper1 is motivated by some recent quantum key distribution (QKD) protocols that make use of continuous quantum states instead of discrete ones. Quantum key distribution (also called quantum cryptogra- phy) allows Alice and Bob to share a secret key that can be used for encrypting messages. Eavesdropping is detectable in such key distribution schemes, as the laws of quantum me- chanics imply that measuring a quantum state generally dis- turbs it. Quantum cryptography uses two channels: a quan- tum channel (e.g., a fiber in which single photons are sent) and a classical public authenticated channel. To share a secret key, a few steps must be performed. First, quantum states are sent from Alice to Bob on the quantum channel. This process gives the two parties correlated random variables, and . Then, Alice and Bob compare a sample of the transmitted in- formation over the public channel. By measuring some appro- priate disturbance metric, they determine an upper bound on the amount of information a possible eavesdropper was able to get, thanks to the laws of quantum mechanics. Finally, they extract a common secret key out of and . The last step of QKD, namely the construction of a common secret key out of correlated random variables is a non-trivial operation. In many QKD schemes such as BB84 (2), and are sim- ply balanced binary random variables, connected by some er- ror probability . In this case, the secret key

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