Efficient Record-and-Replay Arithmetic for Quantum Elliptic-Curve Point Addition
We study reversible secp256k1 point-addition circuits developed through ECDSA.Fail for Shor's elliptic-curve discrete-logarithm algorithm. Two complementary constructions improve record-and-replay GCD arithmetic: Jump-2 groups binary-GCD steps and compresses their decisions using base-5 encoding, while ping-pong uses fixed register alternation and one-bit decisions to avoid full-width comparisons and data-dependent swaps. Fused replay combines doubling and signed addition into one modular correction. Both constructions support quantum-addressed window selection with measurement-based lookup cleanup. We compare three circuits on 100,000 fresh inputs across nine lookup-table configurations. A separately tested repair uses 1,419 qubits and 1.356 million mean executed Toffolis, with no detected failures on another 100,000 inputs. Structured supported-input counterexamples remain, so these tests do not establish all-input correctness. We also provide conditional coherent-error analysis and reversible safegcd comparisons. Under the stated window allowance, repaired-circuit resources lie below Google's low-gate caps and Schrottenloher's low-gate estimates, but differing accounting and correctness evidence preclude formal dominance. The results concern individual window-selected additions, not complete Shor computations.