A Binary Tree Interpretation of Shared Key Generation Using Modular Ananta-Graph Paths

Abstract

This paper presents a formally grounded symmetric key agreement scheme based on modular traversal of Ananta-Graphs, reinterpreted
and visualised through a binary tree model. The underlying encryption protocol is driven by the iterative dynamics of the
Collatz-like transformation $f(n) = (3n + 1) \bmod m$, applied iteratively over a modular graph structure commencing from a shared
public base node. Two communicating parties, Alice and Bob, independently traverse the graph using privately selected iteration
counts, arriving at an identical shared secret without disclosing their private parameters. We introduce the Ananta-Graph Traversal
Inversion Problem (ATIP) and the Ananta-Graph Traversal Distinguishing Problem (ATDP), and formally argue their hardness by reduction
from the Discrete Logarithm Problem (DLP) in a cyclic group setting, as well as through the non-linearity and many-to-one nature of
the modular transformation. The shared traversal endpoint is processed through a hash-based Key Derivation Function (KDF) to obtain
a cryptographically strong session key, decoupling key agreement from key usage and eliminating structural bias. A binary tree abstraction
provides intuitive visualisation of the convergence properties of the scheme. Experimental results demonstrate that prime moduli produce
substantially longer traversal cycles and superior key dispersion, confirming the practical viability of the proposed framework as a
lightweight symmetric key agreement primitive.