Generalizations of CARTS to More Than Two Contexts

Construct natural generalizations of the CARTS rank-transcoding construction that use more than two contexts.

Background

The paper extends the base Calgacus construction from a payload context and a key context to a generalized two-context transform parameterized by contexts a and b and a bijection on rank-vector space. Exact invertibility is proved for this two-context formulation. The authors explicitly identify the construction of natural variants using more than two contexts as unresolved, while restricting the present work to the existing CARTS scheme and its security formalization.

References

It also remains an open question to build natural generalizations allowing for more than two contexts. We leave this task for future work, and focus here on the formalization of the security properties the existing CARTS scheme.

— CARTS: Contextual Autoregressive Rank Transcoding Steganography for Full-Capacity Keyed Text Encoding  (2609.10744 - Ghantous et al., 9 Sep 2026) in Section 3, subsection “Generalized two-context rank transforms”

On the theoretical side, the hardness of the context search problem, the asymptotic structure of collision fibers, the conditions under which commuting key pairs exist, and the design of more general CARTS transforms are all concrete open problems that follow naturally from the framework.

— CARTS: Contextual Autoregressive Rank Transcoding Steganography for Full-Capacity Keyed Text Encoding  (2609.10744 - Ghantous et al., 9 Sep 2026) in Section 7, Conclusion

A primary limitation of the current framework is its restriction to pairwise layer grouping. Extending the optimization to simultaneously share dictionaries across larger groups of layers ($m > 2$) is non-trivial, as the exact closed-form Sylvester solver relies on the simultaneous diagonalization of two matrix pencils, a property that does not analytically generalize to $m$-term systems. Multi-group extensions would necessitate iterative numerical approximations or higher-order tensor factorizations, introducing additional computational overhead and potential numerical instability. Developing scalable, theoretically grounded strategies for $m$-way layer sharing remains a key challenge and constitutes a primary direction for future work.

— GeoPair: Geometry-Preserving Cross-Layer Factorization for Training-Free Transformer Compression  (2609.25963 - Mohammad et al., 22 Sep 2026) in Conclusion and Limitations section