Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time
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Bibliographic record
Abstract
We provide a subexponential algorithm for solving the discrete logarithm problem in Jacobians of high-genus hyperelliptic curves over finite fields. Its expected running time for instances with genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and underlying finite field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g greater-than-or-equal-to theta log q"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mi> ϑ </mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">g \geq \vartheta \log q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a positive constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> ϑ </mml:mi> <mml:annotation encoding="application/x-tex">\vartheta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis e Superscript left-parenthesis StartFraction 5 Over StartRoot 6 EndRoot EndFraction left-parenthesis StartRoot 1 plus StartFraction 3 Over 2 theta EndFraction EndRoot plus StartRoot StartFraction 3 Over 2 theta EndFraction EndRoot right-parenthesis plus o left-parenthesis 1 right-parenthesis right-parenthesis StartRoot left-parenthesis g log q right-parenthesis log left-parenthesis g log q right-parenthesis EndRoot Baseline right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> </mml:mfrac> <mml:mrow> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> ϑ </mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi> ϑ </mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:msqrt> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mi>log</mml:mi> <mml:mo> </mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:msqrt> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O \left ( e^{ \left ( \frac {5}{\sqrt 6} \left ( \sqrt {1 + \frac {3}{2 \vartheta }} + \sqrt {\frac {3}{2 \vartheta }} \right ) + o (1) \right ) \sqrt {(g \log q) \log (g \log q)}} \right ).</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> The algorithm works over any finite field, and its running time does not rely on any unproven assumptions.
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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 | 0.000 | 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 | 0.000 | 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