A polytime proof of correctness of the Rabin-Miller algorithm from Fermat's little theorem
Bibliographic record
Abstract
Although a deterministic polytime algorithm for primality testing is now known, the Rabin-Miller randomized test of primality continues being the most efficient and widely used algorithm. We prove the correctness of the Rabin-Miller algorithm in the theory V1 for polynomial time reasoning, from Fermat's little theorem. This is interesting because the Rabin-Miller algorithm is a polytime randomized algorithm, which runs in the class RP (i.e., the class of polytime Monte-Carlo algorithms), with a sampling space exponential in the length of the binary encoding of the input number. (The class RP contains polytime P.) However, we show how to express the correctness in the language of V1, and we also show that we can prove the formula expressing correctness with polytime reasoning from Fermat's Little theorem, which is generally expected to be independent of V1. Our proof is also conceptually very basic in the sense that we use the extended Euclid's algorithm, for computing greatest common divisors, as the main workhorse of the proof. For example, we make do without proving the Chinese Reminder theorem, which is used in the standard proofs.
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How this classification was reachedexpand
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 | 0.000 | 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 | 0.004 | 0.006 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".