Quantum Algorithm to Identify Division Property of a Multiset

Abstract

Division property-based integral attack is the generalization of integral property developed by blending saturation attack and higher-order differential attack. This attack is considered as a chosen-plaintext attack because the cryptanalyst generates a multiset of plaintext which possesses a certain division property. However, in real-world applications, it is required to find the division property of a given multiset which turns the attack into a known-plaintext attack. The problem, finding the division property of a given multiset $\mathbb {X}$ of size $|\mathbb {X}|$ with each element of n-bit, when solved on a classical computer has the time complexity of $O(n2^n|\mathbb {X}|)$ (fixed in both average and worst cases). In this paper, a better and comparable algorithm using quantum computing is presented along with its quantum oracle designs that can find the correct division property of a multiset in the average case time complexity of $O \left( \log (n)2^n\sqrt{|\mathbb {X}|} \right)$ and worst case time complexity of $O\left( \log (n) 2^n|\mathbb {X}| \right)$ using $\left( n + \lceil \log |\mathbb {X}|\rceil + p \right)$-qubits, where p are the precision qubits required by the quantum counting subroutine.