MILP Modeling for (Large) S-boxes to Optimize Probability of Differential Characteristics
Bibliographic record
Abstract
Current Mixed Integer Linear Programming (MILP)-based search against symmetric-key primitives with 8-bit S-boxes can only build word-wise model to search for truncated differential characteristics. In such a model, the properties of the Differential Distribution Table (DDT) are not considered. To take these properties into account, a bit-wise model is necessary, which can be generated by the H-representation of the convex hull or the logical condition modeling. However, the complexity of both approaches becomes impractical when the size of the S-box exceeds 5 bits. In this paper, we propose a new modeling for large (8-bit or more) S-boxes. In particular, we first propose an algorithm to generate a bit-wise model of the DDT for large S-boxes. We observe that the problem of generating constraints in logical condition modeling can be converted into the problem of minimizing the product-of-sum of Boolean functions, which is a well-studied problem. Hence, classical off-the-shelf solutions such as the Quine-McCluskey algorithm or the Espresso algorithm can be utilized, which makes building a bit-wise model, for 8-bit or larger S-boxes, practical. Then this model is further extended to search for the best differential characteristic by considering the probabilities of each propagation in the DDT, which is a much harder problem than searching for the lower bound on the number of active S-boxes. Our idea is to separate the DDT into multiple tables for each probability and add conditional constraints to control the behavior of these multiple tables. The proposed modeling is first applied to SKINNY-128 to find that there is no differential characteristic having probability higher than 2−128 for 14 rounds, while the designers originally expected that 15 rounds were required. We also applied the proposed modeling to two, arbitrarily selected, constructions of the seven AES round function based constructions proposed in FSE 2016 and managed to improve the lower bound on the number of the active S-boxes in one construction and the upper bound on the differential characteristic for the other.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".