On the distribution of the power generator
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Bibliographic record
Abstract
We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"><mml:semantics><mml:mi>m</mml:mi><mml:annotation encoding="application/x-tex">m</mml:annotation></mml:semantics></mml:math></inline-formula>. This includes as special cases the RSA generator and the Blum–Blum–Shub generator. We prove the uniform distribution of these, provided that the period<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to m Superscript 3 slash 4 plus delta"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>3</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">t\ge m^{3/4 + \delta }</mml:annotation></mml:semantics></mml:math></inline-formula>with fixed<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta greater-than 0"><mml:semantics><mml:mrow><mml:mi>δ</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\delta > 0</mml:annotation></mml:semantics></mml:math></inline-formula>and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to m Superscript 23 slash 24 plus delta"><mml:semantics><mml:mrow><mml:mi>t</mml:mi><mml:mo>≥</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>23</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>24</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding="application/x-tex">t\ge m^{23/24 + \delta }</mml:annotation></mml:semantics></mml:math></inline-formula>. We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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