A Trigonometric Gear-Train Representation of Discrete Logarithm Congruences

This work presents an idealized kinematic representation of the discrete logarithm congruence using a cascade of externally meshing gears. Starting from the modular equation \[g^X \equiv r \pmod p,\] the construction considers a gear train with successive tooth counts \(p, g, g^2, \ldots, g^X\). By applying an initial angular displacement corresponding to \(r\) tooth pitches, together with an integer number of complete revolutions, the angular displacement of the final gear can be expressed in terms of the same integer relation that defines the modular congruence. The central result establishes an exact equivalence: the final gear completes one full revolution precisely when \[r+kp=g^X\] for some nonnegative integer \(k\), which is equivalent to \(g^X\equiv r\pmod p\). The representation is also formulated trigonometrically, using the sine and cosine of the angular positions of the gears. A generalized version is presented for gear trains with arbitrary positive integer tooth counts, extending the construction beyond successive powers of a single base. The purpose of this work is to provide a geometric and physical interpretation of a modular arithmetic relation. It does not propose a new algorithm for solving the discrete logarithm problem, nor does it claim a computational advantage over existing methods. Rather, it develops a particular idealized gear-train model whose kinematics reproduce the underlying modular congruence. The mathematical derivations, assumptions, and relationship between angular phase and accumulated rotation are discussed explicitly.