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Achievable key rates for universal simulation of random data with respect to a set of statistical tests

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1 Author(s)
N. Merhav ; Hewlett-Packard Labs., Palo Alto, CA, USA

We consider the problem of universal simulation of an unknown source from a certain parametric family of discrete memoryless sources, given a training vector X from that source and given a limited budget of purely random key bits. The goal is to generate a sequence of random vectors {Yi}, all of the same dimension and the same probability law as the given training vector X, such that a certain, prescribed set of M statistical tests will be satisfied. In particular, for each statistical test, it is required that for a certain event, ε, 1 ≤ ℓ ≤ M, the relative frequency 1/N Σi=1N 1εℓ(Yi) (1ε(·) being the indicator function of an event ε), would converge, as N → ∞, to a random variable (depending on X), that is typically as close as possible to the expectation of 1εℓ, (X) with respect to the true unknown source, namely, to the probability of the event ε. We characterize the minimum key rate needed for this purpose and demonstrate how this minimum can be approached in principle.

Published in:

IEEE Transactions on Information Theory  (Volume:50 ,  Issue: 1 )