Double-Base Number System for Multi-Scalar Multiplications.
Bibliographic record
Abstract
Abstract. The Joint Sparse Form is currently the standard represen-tation system to perform multi-scalar multiplications of the form [n]P + m[Q]. We introduce the concept of Joint Double-Base Chain, a general-ization of the Double-Base Number System to represent simultaneously n and m. This concept is relevant because of the high redundancy of Double-Base systems, which ensures that we can find a chain of reason-able length that uses exactly the same terms to compute both n and m. Furthermore, we discuss an algorithm to produce such a Joint Double-Base Chain. Because of its simplicity, this algorithm is straightforward to implement, efficient, and also quite easy to analyze. Namely, in our main result we show that the average number of terms in the expansion is less than 0.3945 log2 n. With respect to the Joint Sparse Form, this induces a reduction by more than 20 % of the number of additions. As a consequence, the total number of multiplications required for a scalar multiplications is minimal for our method, across all the methods using two precomputations, P +Q and P −Q. This is the case even with co-ordinate systems offering very cheap doublings, in contrast with recent results on scalar multiplications. Several variants are discussed, including methods using more precomputed points and a generalization relevant for Koblitz curves. Our second contribution is a new way to evaluate φ, the dual endomorphism of the Frobenius. Namely, we propose formulae to compute ±φ(P) with at most 2 multiplications and 2 squarings in F2d. This represents a speed-up of about 50 % with respect to the fastest techniques known. This has very concrete consequences on scalar and multi-scalar multiplications on Koblitz curves.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 0.005 |
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".