Product-Congruence Games: A Unified Impartial-Game Framework for RSA ($\phi$-MuM) and AES (poly-MuM)

TL;DR

This paper introduces a unified impartial-game framework called Product-Congruence Game (PCG) that models RSA and AES cryptographic processes, revealing structural properties and invariants shared by these systems.

Contribution

It develops a novel unified game-theoretic framework for RSA and AES, providing structural theorems and insights into their algebraic cores and invariants.

Findings

Unified framework for RSA and AES via PCG

Structural theorems for general PCG

Analysis of game invariants and collapse phenomena

Abstract

RSA exponent reduction and AES S-box inversion share a hidden commonality: both are governed by the same impartial combinatorial principle, which we call a Product-Congruence Game (PCG). A Product-Congruence Game tracks play via the modular or finite-field product of heap values, providing a single invariant that unifies the algebraic cores of these two ubiquitous symmetric and asymmetric cryptosystems. We instantiate this framework with two companion games. First, $\phi$-MuM, in which a left-associated "multi-secret" RSA exponent chain compresses into the game of Multiplicative Modular Nim, PCG($k,\{1\}$), where $k = ord_N(g)$. The losing predicate then factorizes via the Chinese remainder theorem, mirroring RSA's structure. Second, poly-MuM, our model for finite-field inversion such as the AES S-box. For poly-MuM we prove the single-hole property inside its threshold region, implying that the Sprague-Grundy values are multiplicative under disjunctive sums in that region. Beyond these instances, we establish four structural theorems for a general Product-Congruence Game PCG($m,R$): (i) single-heap repair above the modulus, (ii) ultimate period $m$ per coordinate, (iii) exact and asymptotic losing densities, and (iv) confinement of optimal play to a finite indeterminacy region. An operation-alignment collapse principle explains why some variants degenerate to a single aggregate while MuM, $\phi$-MuM and poly-MuM retain rich local structure. All ingredients (multiplicative orders, the Chinese remainder theorem, finite fields) are classical; the contribution is the unified aggregation-compression viewpoint that embeds both RSA and AES inside one impartial-game framework, together with the structural and collapse theorems.