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Quantum Information and Computation

Efficient quantum circuits for binary elliptic curve arithmetic

AmentoBrittanney, RöttelerMartin, SteinwandtRainer

Published Jul 1, 2013• 0 citations
Publisher

Abstract

Elliptic curves over finite fields F2n play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affi...
DOI
10.5555/2535649.2535654

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Efficient quantum circuits for binary elliptic curve arithmetic: reducing TTT-gate complexity
Elliptic curves over finite fields F2n{\mathbb F}_{2^n}F2n​ play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of TTT-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in F2n\mathbb F_{2^n}F2n​ in depth $\bigO(n\log_2 n)$ using a polynomial basis representation, which may be of independent interest.
Elliptic curves over finite fields GF(2^n) play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with affine or projective coordinates. In this paper we show that changing the curve representation allows a substantial reduction in the number of T-gates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in GF(2^n) in depth O(n log n) using a polynomial basis representation, which may be of independent interest.