Improved Quantum Circuits for Elliptic Curve Discrete Logarithms

• Published in: Post-Quantum Cryptography Vol. 12100; pp. 425 - 444
• Main Authors: Häner, Thomas, Jaques, Samuel, Naehrig, Michael, Roetteler, Martin, Soeken, Mathias
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2020; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Discrete logarithm problem; Elliptic curve cryptography; Quantum cryptanalysis; Resource estimates; Shor’s algorithm
• ISBN: 3030442225; 9783030442224
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-44223-1_23

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.