Another look at some fast modular arithmetic methods
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Bibliographic record
Abstract
Abstract In this work we re-examine a modular multiplication and a modular exponentiation method. The multiplication method, proposed by Hayashi in 1998, uses knowledge of the factorization of both N + 1 and N + 2 to compute a multiplication modulo N . If both N + 1 and N + 2 can be factored into k equally sized relatively prime factors then the computations are done modulo each of the factors and then combined using the Chinese Remainder Theorem. It was suggested that the (asymptotic) computational costs of the method is 1/ k of simply multiplying and reducing modulo N . We show, however, that the computational costs of the method is (asymptotically) at least as costly as simply multiplying and reducing modulo N for both squarings and general multiplications when efficient arithmetic is used. The exponentiation method, proposed by Hwang, Su, Yeh and Chen in 2005, is based on Hayashi's method and uses knowledge of the factorization of P + 1 and P – 1 to compute an exponentiation modulo an odd prime P . We begin by showing that the method cannot be used as a general purpose exponentiation method and then modify the method so that it can work as a general purpose modular multiplication method. Like Hayashi's method, however, this method is at best (asymptotically) only as efficient as simply multiplying and reducing modulo P .
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it