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arXiv:1310.5185 · Quantum Information and Computation

Adversary lower bounds for the collision and the set equality problems

Aleksandrs Belovs, Ansis Rosmanis

Published Mar 1, 2018• 13 citations
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Abstract

We prove tight \Omega(n^{1/3}) 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 negative-weight adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a more recent development by Zhandry.
DOI
10.26421/qic18.3-4-2

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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.
Given a function f as an oracle, the collision problem is to find two distinct inputs i and j such that f(i)=f(j), under the promise that such inputs exist. In this paper, we prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function /spl Omega/ ((n/r)/sup 1/3/) times, where n is the size of the domain and r|n. This lower bound matches, up to a constant factor, the upper bound of Brassard, Hoyer and Tapp (1997), which uses the quantum algorithm of Grover (1996) in a novel way. The previously best quantum lower bound is /spl Omega/ ((n/r)/sup 1/5/) evaluations, due to Aaronson (2002). Our result implies a quantum lower bound of /spl Omega/ (n/sup 2/3/) queries to the inputs for another well studied problem, the element distinctness problem, which is to determine whether or not the given n real numbers are distinct. The previous best lower bound is /spl Omega/ (/spl radic/n) queries in the black-box model; and /spl Omega/ (/spl radic/n log n) comparisons in the comparisons-only model, due to Hoyer Neerbek, and Shi (2001).
The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations interms of weight schemes, eigen values, and Kolmogorov complexity. All these formulations rely on the principlethat if an algorithm successfully computes a function then, in particular, itis able to distinguish between inputs which map to different values.