On the degree of Boolean functions as real polynomials
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
A lower bound for randomized algebraic decision trees
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
On the Power of Quantum Computation
SIAM Journal on Computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
Strengths and Weaknesses of Quantum Computing
SIAM Journal on Computing
Quantum lower bounds by quantum arguments
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Quantum mechanical algorithms for the nonabelian hidden subgroup problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum lower bound for the collision problem
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Quantum Lower Bounds for the Collision and the Element Distinctness Problems
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Quantum Cryptanalysis of Hash and Claw-Free Functions
LATIN '98 Proceedings of the Third Latin American Symposium on Theoretical Informatics
Quantum time-space tradeoffs for sorting
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Quantum Lower Bounds by Polynomials
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Succinct quantum proofs for properties of finite groups
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Quantum Algorithms for Element Distinctness
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Polynomial degree vs. quantum query complexity
Journal of Computer and System Sciences - Special issue on FOCS 2003
The quantum query complexity of the abelian hidden subgroup problem
Theoretical Computer Science
Negative weights make adversaries stronger
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
The pattern matrix method for lower bounds on quantum communication
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Adversary Lower Bounds for Nonadaptive Quantum Algorithms
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
Claw finding algorithms using quantum walk
Theoretical Computer Science
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Quantum random walks - new method for designing quantum algorithms
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Quantum walk based search algorithms
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Adversary lower bounds for nonadaptive quantum algorithms
Journal of Computer and System Sciences
Strong direct product theorems for quantum communication and query complexity
Proceedings of the forty-third annual ACM symposium on Theory of computing
Quantum interpolation of polynomilas
Quantum Information & Computation
Quantum search of partially ordered sets
Quantum Information & Computation
Merkle puzzles in a quantum world
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
SIAM Journal on Computing
A quantum lower bound for the query complexity of simon's problem
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
All quantum adversary methods are equivalent
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Quantum complexity of testing group commutativity
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Promised and distributed quantum search
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Random oracles in a quantum world
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Span programs for functions with constant-sized 1-certificates: extended abstract
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Making polynomials robust to noise
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
The quantum query complexity of AC0
Quantum Information & Computation
Impossibility of succinct quantum proofs for collision-freeness
Quantum Information & Computation
Lossy functions do not amplify well
TCC'12 Proceedings of the 9th international conference on Theory of Cryptography
SIAM Journal on Computing
An improved claw finding algorithm using quantum walk
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Optimal randomized comparison based algorithms for collision
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Adversary lower bound for the k-sum problem
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Time-Efficient quantum walks for 3-distinctness
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Dual lower bounds for approximate degree and markov-bernstein inequalities
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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Given a function f as an oracle, the collision problem is to find two distinct indexes i and j such that f(i) = f(j), under the promise that such indexes exist. Since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions, our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis. We prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function Ω((n/r)1/3) times, where n is the size of the domain and r|n. This matches an upper bound of Brassard, Høyer, and Tapp. No lower bound better than constant was previously known. Our result also implies a quantum lower bound of Ω(n2/3) queries for the element distinctness problem, which is to determine whether n integers are all distinct. The best previous lower bound was Ω(&sqrt;n) queries.