Translation-Randomizable Distributions via Random Walks

This work continues the search for viable intractability assumptions over infinite groups. In particular, we study the possibility of phrasing random self-reducibility properties for infinite groups in an analogous manner to the case of finite groups with the uniform distribution. As a first step, it is natural to look for distributions which are translation-invariant, i.e., the probability of an event and its translate by a group element are the same (as is the case for the uniform distribution). Indeed, this approach has been considered in cryptographic literature by Lee [Lee04], who introduced the concept of right invariance. However, we argue a number of shortcomings for its applicability to cryptography, showing in particular that any computational problem defined on a right-invariant distribution will not yield a better (weaker) intractability assumption than some problem defined over a finite group with the uniform distribution. Perhaps the problem is simply that translation invariance is too strong of a property to ask of a distribution over an infinite group. Any such distribution is necessarily non-atomic, and the atomic approximations introduced by [Lee04] (universally right invariant distributions) are still insufficient to deliver the desired complexity reductions. However, if a family of distributions is randomizable via translation, this may in fact suffice: one could translate an arbitrary instance by a sample from a known distribution, and obtain a related instance distributed according to a desired base distribution (or something statistically close) – highly analogous to the mode of operation of many random self reductions in cryptography. Using a novel approach based on random walks, we construct families of such distributions, which are translation-randomizable over infinite groups. The main ingredients in our construction are recurrence (meaning a random walk will invariably return to its origin), and shortcut sampling, which asserts the existence of an efficient method for sampling a long (super-polynomial length) walk. Given a suitable group with these properties (for instance Z), we demonstrate how one may formulate problems with random self reducibility properties akin to the familiar setting of finite groups and the uniform distribution.