Unbiased bits from sources of weak randomness and probabilistic communication complexity
SIAM Journal on Computing - Special issue on cryptography
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When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits Bj are extracted and combined into the machine number Y := \sum^n_{j=1}2^{-j}B_j. In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables Xn (which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable Un : = X2n − 1/(X2n − 1 + X2n) is uniform in [0, 1]. In the practical application Xn can only be measured up to a given precision ϑ (in terms of the expectation of the Xn); it is shown that the distribution function obtained by calculating Un from these measurements differs from the uniform by less than ϑ/2.We compare this deviation with the error resulting from the use of biased bits Bj with Pϵ{Bj = 1} = \frac{1}{2} + \varepsilon (where ϵ ∈] − \frac{1}{2}, \frac{1}{2}[) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm ‖Q‖TVp = (\sum_{\omega}|Q(ω)|p)1/p (p ≥ 1) we have ‖PϵY − P0Y‖TVp ≤ (cn\sqrt{n} · ϵ)1/p with cn → p \sqrt{8/\pi} for n → ∞. For the distribution function ‖FϵY − F0Y‖ ≤ 2(1 − 2−n)|ϵ| holds.