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Quantum Information and Computation

Adversary lower bounds for the collision and the set equality problems

BelovsAleksandrs, RosmanisAnsis

Published Mar 1, 2018• 0 citations
Publisher

Abstract

We prove tight Ω(n1/3) lower bounds on the quantum query complexity of the Collision and the SET EQUALITY problems, provided that the size of the alphabet is large enough. We do this using the nega...
DOI
10.5555/3370192.3370194

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A collision for a function fff is two distinct inputs x1≠x2x_1\neq x_2x1​=x2​ such that fff outputs the same value on both inputs: f(x1)=f(x2)f(x_1)=f(x_2)f(x1​)=f(x2​). The quantum query complexity of finding collisions has been shown~\cite{BHT1997,AS2004, Ambainis05, Kutin05} in some settings to be Θ(N1/3)\Theta(N^{1/3})Θ(N1/3); however, these results do not apply to random functions. The issues are two-fold. First, the Ω(N1/3)\Omega(N^{1/3})Ω(N1/3) lower bound only applies when the domain is no larger than the co-domain, which precludes many of the cryptographically interesting applications. Second, most of the results in the literature only apply to rrr-to-1 functions, which are quite different from random functions. Understanding the collision problem for random functions is of great importance to cryptography, and we seek to fill the gaps of knowledge for this problem. To that end, we prove that, as expected, a quantum query complexity of Θ(N1/3)\Theta(N^{1/3})Θ(N1/3) holds for all interesting domain and co-domain sizes. Our proofs are simple, and combine existing techniques with several novel tricks to obtain the desired results. Using our techniques, we also give an optimal Ω(M1/3)\Omega(M^{1/3})Ω(M1/3) lower bound for the set equality problem. This lower bound can be used to improve the relationship between classical randomized query complexity and quantum query complexity for so-called permutation-symmetric functions.
The Element Distinctness problem is to decide whether each character of an input string is unique. The quantum query complexity of Element Distinctness is known to be Θ(N2/3)Θ(N^{2/3})Θ(N2/3); the polynomial method gives a tight lower bound for any input alphabet, while a tight adversary construction was only known for alphabets of size Ω(N2)Ω(N^2)Ω(N2). We construct a tight Ω(N2/3)Ω(N^{2/3})Ω(N2/3) adversary lower bound for Element Distinctness with minimal non-trivial alphabet size, which equals the length of the input. This result may help to improve lower bounds for other related query problems.
(MATH) The collision problem is to decide whether a function X: { 1,…,n} → { 1, …,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Ω(n1/5) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O(n1/3), but obtaining any lower bound better than Ω(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Ω(n1/7) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.