Efficient CM-constructions of elliptic curves over finite fields
Bibliographic record
Abstract
We present an algorithm that, on input of an integer N ≥ 1 N\ge 1 together with its prime factorization, constructs a finite field F \mathbf {F} and an elliptic curve E E over F \mathbf {F} for which E ( F ) E({\mathbf {F} }) has order N N . Although it is unproved that this can be done for all N N , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 ω ( N ) log N 2^{\omega (N)}\log N , where ω ( N ) \omega (N) is the number of distinct prime factors of N N . In the cryptographically relevant case where N N is prime, an expected run time O ( ( log N ) 4 + ε ) O((\log N)^{4+\varepsilon }) can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of 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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.007 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.027 | 0.016 |
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".