Minimizing Locality of One-Way Functions via Semi-private Randomized Encodings

A one-way function is d-local if each of its outputs depends on at most d input bits. In Applebaum et al. (SIAM J Comput 36(4):845–888, 2006), it was shown that, under relatively mild assumptions, there exist 4-local one-way functions (OWFs). This result is not far from optimal as it is not hard to show that there are no 2-local OWFs. The gap was partially closed in Applebaum et al. (2006) by showing that the existence of 3-local OWFs is implied by the intractability of decoding a random linear code (or equivalently the hardness of learning parity with noise). In this note we provide further evidence for the existence of 3-local OWFs. We construct a 3-local OWF based on the assumption that a random function of (arbitrarily large) constant locality is one-way. [A closely related assumption was previously made by Goldreich (Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation, pp. 76–87, 2011).] Our proof consists of two steps: (1) we show that, under the above assumption, Random Local Functions remain hard to invert even when some information on the preimage x is leaked and (2) such “robust” local one-way functions can be converted to 3-local one-way functions via a new construction of semi-private randomized encoding. We believe that these results may be of independent interest.