Nimber-Preserving Reduction: Game Secrets and Homomorphic Sprague-Grundy Theorem

The concept of nimbers—a.k.a. Grundy-values or nim-values—is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, I , of polynomially-short impartial rulesets, under polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in I is Sprague-Grundy-complete for I . By viewing every impartial game as an encoding of its nimber—a succinct game secret richer than its winnability alone—our technical result establishes the following striking cryptographyinspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for I , there exists a polynomial-time algorithm to construct, for any pair of games G1, G2 in I , a Generalized Geography game G satisfying: nimber(G) = nimber(G1) ⊕ nimber(G2). Supported by the Simons Investigator Award for fundamental & curiosity-driven research and NSF grant CCF1815254.