Paper 2026/1244

Resource Estimation of the Distributed Quantum Algorithm for the Elliptic Curve Logarithm Problem

Abstract

Elliptic Curve Cryptography (ECC) underpins modern public-key infrastructure, relying on the computational hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). While monolithic quantum architectures running Shor's algorithm threaten ECC, their physical realization is bottlenecked by massive logical qubit demands for modular inversion. Distributed Quantum Computing (DQC) offers a scalable pathway by interconnecting smaller, cooperative Quantum Processing Units (QPUs). In this paper, we adapt two state-of-the-art distributed quantum frameworks, namely a zero-quantum-communication paradigm and a sequential teleportation-based protocol, to the elliptic curve setting. We provide the first rigorous resource estimation of distributed ECDLP solvers. By incorporating the compact, register-sharing Extended Euclidean Algorithm (EEA) formulation, we quantify the physical design trade-offs for a cryptographically secure 256-bit elliptic curve on fault-tolerant architectures. Specifically, the zero-quantum-communication variant operates with 1094 to 1154 logical qubits per node, whereas the teleportation-based variant requires as few as 856 to 1098 logical qubits per node. Furthermore, optimizing the arithmetic window size ($\omega = 16$) over $k = 22$ nodes reduces the single-node Toffoli gate count to $2^{26.84}$, which represents an approximate $14\times$ reduction compared to the $2^{30.63}$ baseline of monolithic schemes. Our results delineate the architectural boundaries between inter-QPU communication, classical coordination, and single-chip hardware floors, mapping out an efficient design space for distributed quantum cryptanalysis.

Note: We updated the quantum resource estimation and complexity analysis by integrating state-of-the-art register-sharing reversible modular arithmetic from Luo et al. (2026).