Period of the power generator and small values of Carmichael’s function
Bibliographic record
Abstract
Consider the pseudorandom number generator un≡un−1e(modm),0≤un≤m−1,n=1,2,…, \begin{equation*} u_n\equiv u_{n-1}^e\pmod {m},\quad 0\le u_n\le m-1,\quad n=1,2,\ldots , \end{equation*} where we are given the modulus m m , the initial value u0=ϑ u_0=\vartheta and the exponent e e . One case of particular interest is when the modulus m m is of the form pl pl , where p,l p,l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m=pl m=pl , if the period of the sequence (un) (u_n) exceeds m3/4+ε m^{3/4+\varepsilon } , then the sequence is uniformly distributed. We show rigorously that for almost all choices of p,l p,l it is the case that for almost all choices of ϑ,e \vartheta ,e , the period of the power generator exceeds (pl)1−ε (pl)^{1-\varepsilon } . And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto. The principal tool is an estimate related to the Carmichael function λ(m) \lambda (m) , the size of the largest cyclic subgroup of the multiplicative group of residues modulo m m . In particular, we show that for any Δ≥(loglogN)3 \Delta \ge (\log \log N)^3 , we have λ(m)≥Nexp(−Δ) \lambda (m)\ge N\exp (-\Delta ) for all integers m m with 1≤m≤N 1\le m\le N , apart from at most Nexp(−0.69(ΔlogΔ)1
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 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".