Fast Software Implementations of Bilinear Pairings
Bibliographic record
Abstract
Advancement in pairing-based protocols has had a major impact on the applicability of cryptography to the solution of more complex real-world problems. However, the computation of pairings in software still needs to be optimized for different platforms including emerging embedded systems and high-performance PCs. Few works in the literature have considered implementations of pairings on the former applications despite their growing importance in a post-PC world. In this paper, we investigate the efficient computation of the Optimal-Ate pairing over special class of pairing friendly Barreto-Naehrig curves in software at different security levels. We target both applications and perform our implementations on ARM-powered processors (with and without NEON instructions) and PC processors. We exploit state-of-the-art techniques and propose new optimizations to speed up the computation in the different levels including tower field and curve arithmetic. In particular, we extend the concept of lazy reduction to inversion in extension fields, analyze an efficient alternative for the sparse multiplication used inside the Miller’s algorithm and reduce further the cost of point/line evaluation formulas in affine and projective homogeneous coordinates. In addition, we study the efficiency of using M-type and D-type sextic twists in the pairing computation and carry out a detailed comparison between affine, Jacobian, and homogeneous coordinate systems. Our implementations on various mass-market emerging embedded devices significantly improve the state-of-the-art of pairing computation on ARM-powered devices and x86-64 PC platforms. For ARM implementations we achieved considerably faster computations in comparison to the counterparts.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".