One-way permutations, computational asymmetry and distortion

TL;DR

This paper explores the concept of computational asymmetry in permutations, introduces related algebraic functions, and connects circuit complexity with group theory, providing new insights into one-way functions and their mathematical properties.

Contribution

It introduces a computational asymmetry function for permutations, relates circuit complexity to Thompson groups, and establishes connections between asymmetry and group distortion functions.

Findings

Circuit size is polynomially equivalent to word-length in Thompson groups.

Circuits with non-fixed-length gates are at most quadratically more compact.

Computational asymmetry relates closely to distortion in Thompson group structures.

Abstract

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.