Quantum Algorithms for Element Distinctness

Abstract

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N ) quantum upper bound for the element distinctness problem in the comparison complexity model (contrasting with Θ(N log N ) classical complexity). We also prove a lower bound of Ω( √ N ) comparisons for this problem and derive bounds for a number of related problems.

Quantum attacks against iterated block ciphers

“…While this may not be a surprise from the point of view of quantum complexity theory (see e.g. the conclusion of [3]), this suggests that the time-space product, a common way of evaluating classical attacks [7], may not be the correct figure of merit to evaluate quantum attacks.…”

“…Interestingly, there exists other algorithms for ED leading to different gains. For example, the algorithm based on amplitude amplification (AA) of [3] leads to a gain in time of 4/3 (over the classical Meet-in-the-middle attack), and a gain in time-space product of 8/5, better than the most time-efficient attack. The most time-efficient algorithm is not the one leading to the most important gain in time-space, and an attacker that is willing to pay with more time can save on the time-space product.…”

A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

“…It would be interesting to determine when one kind of hiding is harder than another. For example, if f is injective save for a single repeated value, then there is a sublinear algorithm for deterministic hiding [6]. But projective hiding requires at least linear time and we do not know an algorithm for coherent hiding which is faster than quadratic time.…”

“…For any given group G, Algorithm 4 requires subalgorithms to compute the character measurement (5) and the tensor decomposition measurement (6). Efficient algorithms for character measurements and character transforms are a topic of active research [4,16] and are unknown for many groups.…”

Quantum Biased Oracles

“…Brassard, et al 11) showed a quantum counting algorithm that gives an approximate counting method by combining the Grover search with the quantum Fourier transformation. Quantum algorithms for the clawfinding and the element distinctness problems given by Buhrman, et al 10) also exploited classical random and sorting methods with Grover Search. (Ambainis,6) developed an optimal quantum algorithm with O(N 2/3 ) queries for element distinctness problem, which makes use of quantum walk and matches to the lower bounds shown by Shi 27) ).…”