Fast and Flexible Elliptic Curve Point Arithmetic over Prime Fields
Bibliographic record
Abstract
We present an innovative methodology for accelerating the elliptic curve point formulas over prime fields. This flexible technique uses the substitution of multiplication with squaring and other cheaper operations by exploiting the fact that field squaring is generally less costly than multiplication. Applying this substitution to the traditional formulas, we obtain faster point operations in unprotected sequential implementations. We also show the significant impact our methodology has in protecting against simple side- channel (SSCA) attacks. We modify the elliptic curve cryptography (ECC) point formulas to achieve a faster atomic structure when applying side-channel atomicity protection. In contrast to previous atomic operations that assume that squarings are indistinguishable from multiplications, our new atomic structure offers true SSCA-protection because it includes squaring in its formulation. Moreover, we extend our implementation to parallel architectures such as Single-Instruction Multiple-Data (SIMD). With the introduction of a new coordinate system and the flexibility of our methodology, we present, to our knowledge, the fastest formulas for SIMD-based schemes that are capable of executing three and four operations simultaneously. Finally, a new parallel SSCA-protected scheme is proposed for multiprocessor/parallel architectures by applying the atomic structure presented in this work. Our parallel and atomic operations are shown to be significantly faster than previous implementations.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".