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

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).

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