Rethinking low genus hyperelliptic Jacobian arithmetic over binary fields: interplay of field arithmetic and explicit formulæ
Bibliographic record
Abstract
Abstract In this paper, we present several improvements on the best known explicit formulæ for hyperelliptic curves of genus three and four in characteristic two, including the issue of reducing memory requirements. To show the effectiveness of these improvements and to allow a fair comparison of the curves of different genera, we implement all formulæ using a highly optimized software library for arithmetic in binary fields. This library was designed to minimize the impact of a whole series of overheads which have a larger significance as the genus of the curves increases. The current state of the art in attacks against the discrete logarithm problem is taken into account for the choice of the field and group sizes. Performance tests are done on two personal computers with very different architectures. Our results can be shortly summarized as follows: Curves of genus three provide performance similar, or better, to that of curves of genus two, and these two types of curves can perform faster than elliptic curves – indeed on some processors often twice as fast. Curves of genus four attain a performance level comparable to elliptic curves. A large choice of curves is therefore available for the deployment of curve-based cryptography, with curves of genus three and four providing their own advantages as larger cofactors can be allowed for the group order.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".