Resolve the principal open tasks in reversible elliptic-curve point addition

Establish canonical replay at competitive cost, justify the value widths and round tails used in reversible ping-pong arithmetic, develop an optimized matched reversible safegcd baseline, and certify the good subspaces and ideal-state weights required for the coherent-error composition bound.

Background

The paper evaluates reversible secp256k1 elliptic-curve point-addition circuits based on record-and-replay modular inversion, including Jump-2 and ping-pong GCD constructions. Although the circuits achieve favorable empirical resource measurements, the authors identify unresolved issues involving the cost of canonical replay, the adequacy of finite round budgets and shrinking register widths, and the lack of a resource-matched reversible safegcd comparison.

The paper also derives a conditional coherent-error composition bound. Applying that bound to a complete Shor computation requires certifying the relevant good input subspaces and determining their weights under the actual ideal intermediate states. The authors explicitly identify these four areas as the principal open tasks.