A framework for reducing the overhead of the quantum oracle for use with Grover's algorithm with applications to cryptanalysis of SIKE release_63ncoc2opre2xo7y2p6cmund2y

by Jean-François Biasse, Benjamin Pring

Published in Journal of Mathematical Cryptology by Walter de Gruyter GmbH.

2020   Volume 15, p143-156

Abstract

<jats:title>Abstract</jats:title>In this paper we provide a framework for applying classical search and preprocessing to quantum oracles for use with Grover's quantum search algorithm in order to lower the quantum circuit-complexity of Grover's algorithm for single-target search problems. This has the effect (for certain problems) of reducing a portion of the polynomial overhead contributed by the implementation cost of quantum oracles and can be used to provide either strict improvements or advantageous trade-offs in circuit-complexity. Our results indicate that it is possible for quantum oracles for certain single-target preimage search problems to reduce the quantum circuit-size from <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jmc-2020-0080_eq_002.png" /><jats:tex-math>$O\left(2^{n/2}\cdot mC\right)$</jats:tex-math></jats:alternatives></jats:inline-formula> (where <jats:italic>C</jats:italic> originates from the cost of implementing the quantum oracle) to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_jmc-2020-0080_eq_003.png" /><jats:tex-math>$O(2^{n/2} \cdot m\sqrt{C})$</jats:tex-math></jats:alternatives></jats:inline-formula> without the use of quantum ram, whilst also slightly reducing the number of required qubits.This framework captures a previous optimisation of Grover's algorithm using preprocessing [21] applied to cryptanalysis, providing new asymptotic analysis. We additionally provide insights and asymptotic improvements on recent cryptanalysis [16] of SIKE [14] via Grover's algorithm, demonstrating that the speedup applies to this attack and impacting upon quantum security estimates [16] incorporated into the SIKE specification [14].
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