Quantum Algorithm for Elliptic Curve Discrete Logarithms with Space-Efficient Point Addition

arXiv:2607.13816 · quant-ph, cs.CR, cs.DS · Submitted 2026-07-15 · Read on arXiv

quant-ph, cs.CR, cs.DS

Submitted: 2026-07-15

Updated: 2026-09-05

Comments: 46 pages, 15 figures, 6 tables. This paper supersedes our earlier preprint arXiv:2604.02311. Compared with the earlier version, the present paper reduces the space complexity from $5n+O(\log_2 n)$ to $3n+O(\log_2 n)$ for affine point addition and from $3n+O(\log_2 n)$ to $2n+O(\log_2 n)$ for modular inversion

Code: https://github.com/ZeroWang030221/Space-Efficient-Quantum-Algorithm-for-Elliptic-CurveDiscrete-Logarithms-with-Resource-Estimation

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Extended Euclidean Algorithm — 7× in this paper
• Quantum Circuits — 7× in this paper · explained in 2 episodes
• Adam — 3× in this paper · explained in 2 episodes
• Core — 3× in this paper · explained in 2 episodes
• Ancilla Qubits — 2× in this paper · explained in 1 episode
• Elliptic Curve Discrete Logarithm Problem — 2× in this paper
• Fidelity — 2× in this paper · explained in 2 episodes
• Transformer — 2× in this paper · explained in 2 episodes
• Abstraction — 1× in this paper · explained in 1 episode
• Decoded Quantum Interferometry — 1× in this paper
• Decoded Quantum Interferometry (DQI) — 1× in this paper · explained in 2 episodes
• Homogeneity — 1× in this paper · explained in 2 episodes
• Trace — 1× in this paper · explained in 2 episodes
• Verification — 1× in this paper · explained in 2 episodes

Sources

• Space-Efficient Quantum Algorithm for Elliptic Curve Discrete Logarithms with Resource Estimation
• Securing Elliptic Curve Cryptocurrencies against Quantum Vulnerabilities: Resource Estimates and Mitigations
• A new quantum ripple-carry addition circuit
• Revisiting Shor's quantum algorithm for computing general discrete logarithms
• How to factor 2048 bit RSA integers with less than a million noisy qubits
• Resource analysis of Shor's elliptic curve algorithm with an improved quantum adder on a two-dimensional lattice
• Verifiable Quantum Advantage via Optimized DQI Circuits
• A Classical-Quantum Adder with Constant Workspace and Linear Gates
• Optimized Point Addition Circuits for Elliptic Curve Discrete Logarithms
• Measurement-based Uncomputation Applied to Controlled Modular Multiplication
• AlphaEvolve: A coding agent for scientific and algorithmic discovery
• Structured Scaling of AI Discovery Across Diverse Scientific Domains

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