An O(m2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2m)a

Authors:
Dmitri Maslov;Jimson Mathew;Donny Cheung;Dhiraj K. Pradhan
Affiliations:
Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada and Directorate for Computer and Information Science & Engineering, National Science Foundation, Arlingto ...;Department of Computer Science, University of Bristol, Bristol, UK;Department of Computer Science, University of Calgary, Calgary, Alberta, Canada;Department of Computer Science, University of Bristol, Bristol, UK
Venue:
Quantum Information & Computation
Year:
2009

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Abstract

We consider a quantum polynomial-time algorithm which solves the discrete logarithmproblem for points on elliptic curves over GF(2m). We improve over earlier algorithmsby constructing an efficient circuit for multiplying elements of binary finite fields andby representing elliptic curve points using a technique based on projective coordinates.The depth of our proposed implementation, executable in the Linear Nearest Neighbor(LNN) architecture, is O(m2), which is an improvement over the previous bound ofO(m3) derived assuming no architectural restrictions.