Steganographic information hiding via symmetric numerical semigroups

TL;DR

The paper addresses covert communication by hiding information in the gap structure of symmetric numerical semigroups derived from the Frobenius coin problem. It proposes a modular gap-partitioning encoding scheme, where data are mapped to gaps with prescribed residues modulo a small modulus $M$ and decoded efficiently via a residue-graph membership test enabled by a private telescopic generating set. Key contributions include formalizing the encoding/decoding framework, analyzing indistinguishability granted by symmetry (gap density $g(\mathcal{S}) = (F(\mathcal{S})+1)/2$), and introducing salting to widen the observable range while preserving decodability; alongside a discussion of average-case hardness for gap-inference and resistance to lattice-based attacks. The approach offers a novel, post-quantum resilient number-theoretic primitive for covert communication, connecting additive combinatorics with steganography and providing a foundation for further empirical indistinguishability testing and parameter tuning.

Abstract

We introduce a steganographic information hiding scheme based on structural properties of numerical semigroups arising from the Frobenius coin problem. Instead of encoding data through representable integers, the proposed protocol embeds information into the gap structure of carefully chosen symmetric numerical semigroups. Symmetry guarantees a balanced gap density, ensuring that encoded values are statistically indistinguishable from uniform numerical noise to an observer lacking the private generating set. The security of the scheme relies on the assumed average-case hardness of numerical semigroup membership inference for hidden generators, offering a novel number-theoretic primitive for covert communication and post-quantum resilient information hiding.