Slope packings and coverings, and generic algorithms for the discrete logarithm problem
Bibliographic record
Abstract
Abstract We consider the set of slopes of lines formed by joining all pairs of points in some subset S of a Desarguesian affine plane of prime order p. If all the slopes are distinct and non‐infinite, we have a slope packing; if every possible non‐infinite slope occurs, then we have a slope covering. We review and unify some results on these problems that can be derived from the study of Sidon sets and sum covers. Then we report some computational results, we have obtained for small values of p. Finally, we point out some connections between slope packings and coverings and generic algorithms for the discrete logarithm problem in prime order (sub)groups. Our results provide a combinatorial characterization of such algorithms, in the sense that any generic algorithm implies the existence of a certain slope packing or covering, and conversely. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 36–50, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10033
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".