Resource analysis of Shor's elliptic curve algorithm with an improved quantum adder on a two-dimensional lattice

Abstract: Quantum computers have the potential to break classical cryptographic systems by efficiently solving problems such as the elliptic curve discrete logarithm problem using Shor's algorithm. While resource estimates for factoring-based cryptanalysis are well established, comparable evaluations for Shor's elliptic curve algorithm under realistic architectural constraints remain limited. In this work, we propose a carry-lookahead quantum adder that achieves Toffoli depth $\log n + \log\log n + O(1)$ with only $O(n)$ ancillas, matching state-of-the-art performance in depth while avoiding the prohibitive $O(n\log n)$ space overhead of existing approaches. Importantly, our design is naturally compatible with the two-dimensional nearest-neighbor architectures and introduce only a constant-factor overhead. Further, we perform a comprehensive resource analysis of Shor's elliptic curve algorithm on two-dimensional lattices using the improved adder. By leveraging dynamic circuit techniques with mid-circuit measurements and classically controlled operations, our construction incorporates the windowed method, Montgomery representation, and quantum tables, and substantially reduces the overhead of long-range gates. For cryptographically relevant parameters, we provide precise resource estimates. In particular, breaking the NIST P-256 curve, which underlies most modern public-key infrastructures and the security of Bitcoin, requires about $4300$ logical qubits and logical Toffoli fidelity about $10^{-9}$. These results establish new benchmarks for efficient quantum arithmetic and provide concrete guidance toward the experimental realization of Shor's elliptic curve algorithm.