Parallelism strategies for the tuneable golden-claw finding problem

In this paper we study a strategy for adapting the “Tiny Claw” Grover-based attack of Biasse and Pring (A framework for reducing the overhead of the quantum oracle for use with Grover's algorithm with applications to cryptanalysis of SIKE, J. Math. Cryptol. 15 (2019), pp. 143–156) for attacking SIKE and abstract it under a realistic model of classical memory-access costs. Our results allow us to retain the almost quadratic reduction in the overheads involved with the implementing the quantum oracle in this cost model and demonstrate how the cost of the parallel version of this attack scales in a manner superior to that of a naive use of Grover’s algorithm. In order to investigate the utility of the Tiny Claw approach, we perform a quantum resource estimation of the classical and quantum resources required to attack various SIKE instances with Tiny Claw when when we are limited to hardware, finding interesting price-points.