A Tight Computational Indistinguishability Bound for Product Distributions

Abstract

Assume that distributions $X_0,X_1$ (respectively $Y_0,Y_1$) are $d_X$ (respectively $d_Y$) indistinguishable for circuits of a given size. It is well known that the product distributions $X_0Y_0,\,X_1Y_1$ are $d_X+d_Y$ indistinguishable for slightly smaller circuits. However, in probability theory where unbounded adversaries are considered through statistical distance, it is folklore knowledge that in fact $X_0Y_0$ and $X_1Y_1$ are $d_X+d_Y-d_X\cdot d_Y$ indistinguishable, and also that this bound is tight.

We formulate and prove the computational analog of this tight bound. Our proof is entirely different from the proof in the statistical case, which is non-constructive. As a corollary, we show that if X and Y are d indistinguishable, then k independent copies of X and k independent copies of Y are almost $1-(1-d)^k$ indistinguishable for smaller circuits, as against $d\cdot k$ using the looser bound.

Our bounds are useful in settings where only weak (i.e. non-negligible) indistinguishability is guaranteed. We demonstrate this in the context of cryptography, showing that our bounds, coupled with the XOR Lemma, yield straightforward computational generalization to the analysis for information-theoretic amplification of weak oblivious transfer protocols.

Supported by the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No. 101042417, acronym SPP), by ISF grant 18/484, and by Len Blavatnik and the Blavatnik Family Foundation.