Concrete quantum cryptanalysis of binary elliptic curves

  • Gustavo Banegas Chalmers University of Technology, Gothenburg, Sweden
  • Daniel J. Bernstein University of Illinois at Chicago, Chicago, USA; Ruhr University Bochum, Bochum, Germany
  • Iggy van Hoof Eindhoven University of Technology, Eindhoven, The Netherlands
  • Tanja Lange Eindhoven University of Technology, Eindhoven, The Netherlands
Keywords: Quantum cryptanalysis, elliptic curves, quantum resource estimation, quantum gates, Shor’s algorithm

Abstract

This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2(n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2(n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography.

Published
2020-12-03
How to Cite
Banegas, G., Bernstein, D. J., van Hoof, I., & Lange, T. (2020). Concrete quantum cryptanalysis of binary elliptic curves. IACR Transactions on Cryptographic Hardware and Embedded Systems, 2021(1), 451-472. https://doi.org/10.46586/tches.v2021.i1.451-472
Section
Articles