Small CM discriminant and ECDLP security

Formal statement

Let $E/\mathbb F_p$ be ordinary with complex multiplication by an order of discriminant $D$, where $|D|$ is small.

Prove a polynomial-time reduction from ECDLP on small-$|D|$ curves to ECDLP on general curves, or give a subexponential ECDLP algorithm exploiting $|D|$.

Initial empirical route

1. Construct and independently count curves for several small discriminants.
2. Implement Pollard rho modulo an efficiently computable endomorphism action.
3. Measure the speedup and separate constant factors from asymptotic effects.
4. Treat $D=-3$ and $D=-4$ separately because of their extra automorphisms.

Validation targets

• Independently verify every constructed curve order.
• Verify the endomorphism equation on sampled points.
• Recover known discrete logarithms with both baseline and quotient walks.
• Record operation counts, effect sizes, sample counts, and confidence intervals.

Scope

Only toy parameters with $\log_2 p\leq60$ are in scope.

State in five lines

The project is marked failed/abandoned at the user's explicit request. Sessions 1-3 remain preserved as validated partial research. The $D=-3,-4$ unit experiments and $D=-7$ implementation/tests passed. A003 and A004 establish stated-model obstructions for non-unit quotient rho. A005 stopped before implementation, so Q025 remains completely untested.

What is established (tagged)

• EMPIRICAL: 3,200 recovered DLPs Unit-orbit collision-table speedups matched the fixed $\sqrt6$ and $2$ predictions within all recorded 95% intervals.
• EMPIRICAL: five curves, p=977..262007 The explicit $D=-7$ map passed independent counts and characteristic-equation checks.
• EMPIRICAL: r=233..32831 The measured eigenvalue orders were 116, 495, 1352, 8069, and 16415.
• PROVED Least-label normalization needs $m-1$ successor queries in the sequential successor/comparison model.
• PROVED Addition does not descend to nontrivial scalar-orbit classes.
• PROVED A multiplier-returning canonicalizer with quotient size $q$ reduces ECDLP to $q+1$ canonicalizer calls.

Failure record

• PROVED The initial direct-path unittest invocation failed before discovery because the problem directory contains a dot.
• PROVED A diagnostic used secret 37 in an order-29 subgroup and initially misread the correct residue 8 as an error.
• PROVED A005 was terminated before implementation when the user requested that the work be marked as a failure.
• PROVED The final failure designation is procedural; it is not a mathematical refutation of the preserved results or of Q025.

What remains open

Q025 asks whether several unoriented subgroup-membership constraints can be combined in sub-square-root time. No A005 code or data exist.

Resume condition

Resume only if the user explicitly reopens Q025. Start from the preregistered plan in attempts/A005-unoriented-constraints.md.

Files that matter

• STATE.md and LOG.md: failure/abandonment record.
• attempts/A005-unoriented-constraints.md: stopped plan and post-mortem.
• code/cm_nonunit.py, code/measure_nonunit_orbits.py: validated earlier work.
• data/measure_nonunit_orbits_b10-12-14-16-18_n16_s72022026_20260711_*.csv: preserved evidence.

What I would tell my replacement

Do not interpret the final failure marker as failed mathematics. It records the user's instruction to stop before A005 was implemented.