Input Locality and Hardness Amplification

Abstract

We establish new hardness amplification results for one-way functions in which each input bit influences only a small number of output bits (a.k.a. input-local functions). Our transformations differ from previous ones in that they approximately preserve input locality and at the same time retain the input size of the original function.

Let f : {0, 1} n → {0, 1} m be a one-way function with input locality d, and suppose that f cannot be inverted in time \(\exp(\tilde{O}(\sqrt{n}\cdot d))\) on an ε-fraction of inputs. Our main results can be summarized as follows:

  • If f is injective then it is equally hard to invert f on a (1 − ε)-fraction of inputs.

  • If f is regular then there is a function g: {0, 1} n → {0, 1} m + O(n) that is d + O(log3 n) input local and is equally hard to invert on a (1 − ε)-fraction of inputs.

A natural candidate for a function with small input locality and for which no sub-exponential time attacks are known is Goldreich’s one-way function. To make our results applicable to this function, we prove that when its input locality is set to be d = O(logn) certain variants of the function are (almost) regular with high probability.

In some cases, our techniques are applicable even when the input locality is not small. We demonstrate this by extending our first main result to one-way functions of the “parity with noise” type.