Secret key generation for correlated Gaussian sources

Abstract

We consider secret key (SK) generation by two terminals, each of which observes an i.i.d. source which is correlated with the other; the two sources are jointly Gaussian with known distribution. The terminals are then allowed to communicate with each other, possibly interactively in many rounds and without rate restrictions, over a noiseless two-way public channel. Randomization is permitted at each terminal. The goal is for the terminals to generate a SK comprising discrete common randomness which is concealed from an eavesdropper that has access to the public interterminal communication. We establish that this maximum rate of SK generation, i.e., SK capacity, is, as expected, the per-symbol mutual information I of the correlated sources. Our main technical contribution is a new scheme for achieving SK capacity using structured codes. In our scheme, vector quantization at rate R nats/source symbol of the source at terminal 1 is performed by employing nested lattice codes with dithering. Then, a SK is generated by the terminals from the quantized random sequence at terminal 1 and the (unquantized) Gaussian source at terminal 2, using both lossy and lossless data compression techniques. Our algorithm, using a rate-R lattice quantizer, achieves a maximum SK rate of 1/2 log 1/(e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2I</sup> +(1-e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2I</sup> )e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2R</sup> ), which tends to the SK capacity I with increasing R. Thus, our result also specifies the associated tradeoff between quantization rate R and the maximum achievable SK rate. This tradeoff is shown to be optimum among a certain restricted class of schemes for SK generation that involve quantization at rate R of a Gaussian source at one terminal.