Cryptology ePrint Archive: Report 2020/077

Improved Quantum Circuits for Elliptic Curve Discrete Logarithms

Thomas Häner and Samuel Jaques and Michael Naehrig and Martin Roetteler and Mathias Soeken

Abstract: We present improved quantum circuits for elliptic curve scalar multiplication, the most costly component in Shor's algorithm to compute discrete logarithms in elliptic curve groups. We optimize low-level components such as reversible integer and modular arithmetic through windowing techniques and more adaptive placement of uncomputing steps, and improve over previous quantum circuits for modular inversion by reformulating the binary Euclidean algorithm. Overall, we obtain an affine Weierstrass point addition circuit that has lower depth and uses fewer T gates than previous circuits. While previous work mostly focuses on minimizing the total number of qubits, we present various trade-offs between different cost metrics including the number of qubits, circuit depth and T-gate count. Finally, we provide a full implementation of point addition in the Q# quantum programming language that allows unit tests and automatic quantum resource estimation for all components.

Category / Keywords: public-key cryptography / discrete logarithm problem, quantum cryptanalysis, Shor's algorithm, resource estimates

Original Publication (with major differences): PQCrypto 2020

Date: received 24 Jan 2020

Contact author: samuel jaques at materials ox ac uk,mnaehrig@microsoft com

Available format(s): PDF | BibTeX Citation

Version: 20200126:194353 (All versions of this report)

Short URL: ia.cr/2020/077


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