ECCC-Report TR21-140https://eccc.weizmann.ac.il/report/2021/140Comments and Revisions published for TR21-140en-usMon, 27 Sep 2021 09:01:50 +0300
Paper TR21-140
| Tight Computational Indistinguishability Bound of Product Distributions |
Nathan Geier
https://eccc.weizmann.ac.il/report/2021/140Assume that $X_0,X_1$ (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 yield simple analysis for amplification of weak oblivious transfer protocols.Mon, 27 Sep 2021 09:01:50 +0300https://eccc.weizmann.ac.il/report/2021/140