math.st

Open problems in elliptic-curve cryptography, worked by machine intelligence.

One paper per problem. Every claim carries an epistemic tag. Negative results are kept, not hidden.

This repository is the working record of the program: for each problem, its formal statement, everything established so far, the attempts that failed, and the code and data behind every empirical claim.

What's here

problems/     21 open problems, grouped by category (see below)
papers/       one write-up per problem (.typ source + compiled .pdf)
lib/          shared library — curves, pairings, isogenies, dlog, finite fields
env/          toolchain notes and environment checks
BIBLIOGRAPHY.md · GLOSSARY.md · OPEN_QUESTIONS.md · 00_SCAFFOLD.md

Each problem folder holds:

file	what it is
PROBLEM.md	the formal statement
STATE.md	current status, one-line state, last/next action
LOG.md	session-by-session record (predictions written before running)
HANDOFF.md	findings and state of play for the next session
attempts/	individual attempts, including dead ends, with post-mortems
code/ · data/ · refs/	scripts, their outputs, and checked sources

Categories

1. Hardness of the discrete logarithm — lower bounds, factor bases, descent and transfers
2. Assumptions & reductions — what the standard assumptions actually buy, and their limits
3. Isogenies & quantum — endomorphism rings, class-group actions, torsion leakage
4. Pairings & exTNFS — pairing-friendly families and cycles, extended NFS cost
5. Curve generation & practice — Koblitz-type conjectures, CM security, hash-to-curve

Status

21 problems: 3 resolved, 1 complete (within its stated scope), 12 partial (still open), 1 failed, 4 abandoned. A problem is only marked resolved when the result survives a second, adversarial self-verification; work that stalls or gets ruled out is recorded as such.

Epistemic tags

Claims are tagged inline so the strength of each statement is explicit:

[PROVED] · [REFUTED] · [CITED] · [EMPIRICAL] · [CONDITIONAL] · [HEURISTIC] · [CONJECTURE] · [UNVERIFIED]

A theorem and a toy-scale observation should never read the same, so they don't.

Reproducing

Empirical claims name the script and dataset that produced them (code/<script>.py → data/<name>_<date>.csv). The shared library runs on CPython with no heavyweight CAS dependency; where SageMath/PARI/Magma were unavailable, exact arithmetic fallbacks were used and are noted in env/SETUP.md.