Modular Ackermann maps and hierarchical hash constructions

TL;DR

This paper explores modular Ackermann maps on finite rings, analyzing their structural properties and proposing hierarchical hash constructions based on their depth-dependent behavior, along with conjectures on their distributions and cycle structures.

Contribution

It introduces a novel hierarchy of modular Ackermann maps and develops new hash constructions based on their properties, along with posing open problems.

Findings

Hierarchical structure of Ackermann maps with increasing complexity

Distribution and cycle structure insights for these maps

Open conjectures on asymptotic behavior and properties

Abstract

We introduce and study modular truncations of the Ackermann function viewed as self-maps on finite rings. These maps form a hierarchy of rapidly increasing compositional complexity indexed by recursion depth. We investigate their structural properties, sensitivity to depth variation, and induced distributions modulo powers of two. Motivated by these properties, we define hierarchical hash-type constructions based on depth-dependent Ackermann evaluation. Several conjectures and open problems on distribution, cycle structure, and asymptotic behavior are proposed.