DocumentCode :
390736
Title :
Quantum lower bounds for the collision and the element distinctness problems
Author :
Shi, Yaoyun
Author_Institution :
Dept. of Electr. Eng. & Comput. Sci., Michigan Univ., Ann Arbor, MI, USA
fYear :
2002
fDate :
2002
Firstpage :
513
Lastpage :
519
Abstract :
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 Ω ((n/r)13/) 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 Ω ((n/r)15/) evaluations, due to Aaronson (2002). Our result implies a quantum lower bound of Ω (n23/) 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 Ω (√n) queries in the black-box model; and Ω (√n log n) comparisons in the comparisons-only model, due to Hoyer Neerbek, and Shi (2001).
Keywords :
computational complexity; quantum computing; quantum cryptography; collision problem; cryptography; integer factorization; oracle; quantum algorithm; quantum cryptanalysis; quantum lower bound; Algorithm design and analysis; Computer science; Cryptography; Quantum computing; Quantum mechanics; Upper bound;
fLanguage :
English
Publisher :
ieee
Conference_Titel :
Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on
ISSN :
0272-5428
Print_ISBN :
0-7695-1822-2
Type :
conf
DOI :
10.1109/SFCS.2002.1181975
Filename :
1181975
Link To Document :
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