Computationally Simple One-Way Functions

An upcoming FOCS paper is drawing a lot of interest at TTI and Chicago, Cryptography in NC⁰ by Benny Applebaum, Yuval Ishai and Eyal Kushilevitz.

Don't let "NC⁰" scare you, it simply means that every output bit depends on a constant number of input bits. In this paper the authors show that under reasonable assumptions one can have one-way functions and pseudo-random generators where each output bit depends on only four input bits. Quite surprising that such simple functions can be so hard to invert.

Their proof uses a clever but not overly complicated reduction using randomized polynomials from a larger class (⊕L/poly) to NC⁰ and so if one-way function and PRGs exist in the larger class than slightly weaker versions exist in NC⁰ respectively. The larger class contains many functions conjectured to be one-way or a PRG, including those based on factoring.

I remember Mike Sipser mentioning in a complexity class I took back in the mid-80's that "We don't even know how to show there are no one-way functions in NC⁰." Now we know why.