On the Design and Optimization of a Quantum Polynomial-Time Attack on Elliptic Curve Cryptography

Authors:
Donny Cheung;Dmitri Maslov;Jimson Mathew;Dhiraj K. Pradhan
Affiliations:
Department of Computer Science, and Institute for Quantum Information Science, University of Calgary, Calgary, Canada T2N 1N4;Department of Combinatorics and Optimization, and Institute for Quantum Computing, University of Waterloo, Waterloo, Canada N2L 3G1;Department of Computer Science, University of Bristol, Bristol, UK BS8 1UB;Department of Computer Science, University of Bristol, Bristol, UK BS8 1UB
Venue:
Theory of Quantum Computation, Communication, and Cryptography
Year:
2008

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Abstract

We consider a quantum polynomial-time algorithm which solves the discrete logarithm problem for points on elliptic curves over GF (2 m ). We improve over earlier algorithms by constructing an efficient circuit for multiplying elements of binary finite fields and by representing elliptic curve points using a technique based on projective coordinates. The depth of our proposed implementation is O (m 2), which is an improvement over the previous bound of O (m 3).