# Quantum lower bound for the collision problem

@inproceedings{Aaronson2002QuantumLB,
title={Quantum lower bound for the collision problem},
author={Scott Aaronson},
booktitle={STOC '02},
year={2002}
}
• S. Aaronson
• Published in STOC '02 19 November 2001
• Mathematics, Physics, Computer Science
(MATH) The collision problem is to decide whether a function X: { 1,&ldots;,n} → { 1, &ldots;,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…
124 Citations

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## References

SHOWING 1-10 OF 44 REFERENCES
Quantum lower bounds by polynomials
• Mathematics, Computer Science
JACM
• 2001
This work examines the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1}N in the black-box model and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.
Quantum lower bounds for the set equality problems
The set equality problem is to decide whether two sets $A$ and $B$ are equal or disjoint, under the promise that one of these is the case. Some other problems, like the Graph Isomorphism problem, is
Quantum lower bounds by quantum arguments
Two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques are proved.
Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables
• Yaoyun Shi
• Mathematics, Physics
Inf. Process. Lett.
• 2000
Abstract We prove that, to compute a total Boolean function f on N variables with bounded error probability, any quantum black-box algorithm has to query at least Ω(ρ f N)=Ω( S f ) times, where ρ f
A quantum lower bound for the collision problem
We extend Shi's 2002 quantum lower bound for collision in $r$-to-one functions with $n$ inputs. Shi's bound of $\Omega((n/r)^{1/3})$ is tight, but his proof applies only in the case where the range
Succinct quantum proofs for properties of finite groups
• J. Watrous
• Mathematics, Computer Science
Proceedings 41st Annual Symposium on Foundations of Computer Science
• 2000
It is proved that for an arbitrary group oracle, there exist succinct (polynomial-length) quantum proofs for the Group Non-Membership problem that can be checked with small error in polynomial time on a quantum computer.
Quantum Cryptanalysis of Hash and Claw-Free Functions
• Mathematics, Computer Science
LATIN
• 1998
We give a quantum algorithm that finds collisions in arbitrary r-to-one functions after only O(3√N/r) expected evaluations of the function, where N is the cardinality of the domain. Assuming the
A fast quantum mechanical algorithm for database search
In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .
Quantum lower bounds for the collision and the element distinctness problems
• Mathematics, Computer Science
JACM
• 2004
These lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis, and implies a quantum lower bound of Ω(<i>n</i><sup>2/3</sup>) queries for the element distinctness problem, which is to determine whether <i*n> integers are all distinct.
Algorithms for quantum computation: discrete logarithms and factoring
• P. Shor
• Mathematics, Computer Science
Proceedings 35th Annual Symposium on Foundations of Computer Science
• 1994
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.