Automated Models Improved Cryptographic Divisibility Bounds
New modeling techniques identified tighter divisibility properties for several ciphers, enabling more rigorous security assessments.
Updated on Oct. 14, 2026 in Mathematics
Researchers have developed automated MILP/SAT models that establish improved divisibility bounds for cryptographic ciphers including SIMON, PRESENT, and SPECK. These research-stage methods provide more precise analysis by using complete-round Walsh coefficients.
Why it matters
The research addresses significant gaps in cipher analysis where previous trail-existence tests failed to account for complex dependencies. By improving these bounds, cryptanalysts can more accurately assess the structural security of lightweight and standard cryptographic algorithms.
The models achieve higher divisibility bounds for 674 cases across the SIMON family, notably increasing 9-round SIMON32 bounds from 2^5 to 2^11 and 10-round SIMON48 bounds from 2^6 to 2^22. These improvements utilize complete-round Walsh coefficients and ultrametric representations.
The details
The process uses automated Mixed-Integer Linear Programming (MILP) and Satisfiability (SAT) solvers to perform two-way conversions between linear, integral, and differential-linear sums. By analyzing complete-round Walsh coefficients — mathematical values representing the correlation between inputs and outputs in a function — the models identify zeros and divisibility properties more effectively than traditional trail-existence tests. This approach also employs fast Walsh-Hadamard transforms to systematically account for output condition dependencies.
Timeline
The research paper was published in 2026.
The Tech Race
This research follows a pattern set by the development of integral and differential-linear cryptanalysis, extending the rigor of these established analytical frameworks. It shifts the competitive landscape by automating the identification of cipher vulnerabilities that previously escaped standard trail-existence tests.
While these findings are currently confined to the research community, they inform the security standards used in future cryptographic implementation. Designers and security auditors will eventually use these automated models to verify the robustness of new algorithms against advanced cryptanalytic attacks.
The takeaway
The research highlights how automated formal verification can bridge gaps in cryptanalysis where manual tests fall short. Practitioners should monitor future updates to these SAT-based models to see how they further redefine security bounds for ciphers like GIFT-64 and Ascon.
Further reading
For more information on the current methods used to analyze cipher security, visit the Mathematics section.
More information
Access the full cryptanalytic research paper to review the complete methodology and results.
