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

2026-07-15

AI Summary

This work addresses the quantum solution of the elliptic curve discrete logarithm problem (ECDLP) by proposing a space-efficient reversible modular inversion circuit. Built upon the extended Euclidean algorithm and incorporating mid-circuit measurement with classical feedforward, the design enables controlled point addition in affine coordinates. The approach innovatively introduces a length register and a register-sharing technique for position-controlled arithmetic, substantially reducing qubit overhead. When applied to a 256-bit prime-field elliptic curve, the construction requires only 835 logical qubits—representing a 24%–29% reduction compared to the previous best-known implementation—while maintaining Toffoli gate complexity at approximately $919n^3 / \log_2 n$.

Abstract

The Elliptic Curve Discrete Logarithm Problem (ECDLP) is a fundamental problem in cryptography, and reducing the resource requirements of quantum algorithms for solving ECDLP is an important goal. In this work, we present a space-efficient quantum algorithm for solving the ECDLP over prime fields, achieving an implementation with only $3n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $919n^3/\log_2 n+O(n^2)$ Toffoli gates, where $n$ is the bit-length of the prime. For a 256-bit prime-field curve, our construction requires only 835 logical qubits, reducing the previous best estimates of 1098 and 1175 logical qubits by Chevignard et al. [EUROCRYPT 2026] and Babbush et al. [ArXiv Preprint 2026], respectively. The key to our improvement is a new space-efficient reversible modular inversion circuit, which addresses the dominant space bottleneck in affine-coordinate point addition. Starting from the extended Euclidean algorithm (EEA), we refine the register-sharing technique of Proos and Zalka by introducing length registers and location-controlled arithmetic to compactly store and update intermediate variables. We further optimize the reversible update procedures and construct the corresponding controlled arithmetic circuits, resulting in a modular inversion circuit implemented by only $2n+6\lfloor \log_2 n \rfloor+O(1)$ logical qubits and $195n^2+O(n\log_2 n)$ Toffoli gates. This modular inversion circuit together with mid-circuit measurements and classical feed-forward operations provides a space-efficient controlled affine point-addition circuit and a complete implementation of Shor's algorithm for ECDLP.

Problem

• Elliptic Curve Discrete Logarithm Problem
• quantum algorithm
• space efficiency
• logical qubits
• modular inversion

Innovation

• quantum algorithm
• elliptic curve discrete logarithm problem
• space-efficient
• modular inversion
• reversible computing