Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modular Ackermann maps and hierarchical hash constructions

Jean-Christophe Pain

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.

Modular Ackermann maps and hierarchical hash constructions

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.

Paper Structure

This paper contains 19 sections, 5 theorems, 28 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The modular Ackermann hierarchy
  3. Structural hierarchy
  4. Growth and nonlinear amplification
  5. Distribution modulo powers of two
  6. Cycle structure
  7. Depth-dependent Ackermann constructions
  8. Security and practical considerations
  9. Variants
  10. Dual-depth mixing
  11. Iterated depth hierarchy
  12. Experimental results
  13. Measured statistics
  14. Interpretation and empirical conclusion
  15. Tetration modulo powers of two and hierarchy mixing
  16. ...and 4 more sections

Key Result

Proposition 2.3

For each $m\ge0$, the maps $A_N(m,\cdot)$ satisfy for $n>0$.

Figures (1)

  • Figure 1: Illustration of cycles modulo $2^k$. Red nodes/arrows: exponential dynamics ($m=3$); black nodes/arrows: tetration dynamics ($m=4$). The tetration cycle covers more residues and exhibits stronger mixing.

Theorems & Definitions (21)

  • Definition 2.1: Modular Ackermann map
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: Discontinuity across depth
  • proof
  • Remark 2.5: Heuristic explanation
  • Proposition 2.6
  • Remark 2.7: Empirical distribution observations
  • Conjecture 2.8: Asymptotic equidistribution
  • ...and 11 more