Quantum Algorithm for the Collision Problem

@inproceedings{Brassard2016QuantumAF,
  title={Quantum Algorithm for the Collision Problem},
  author={Gilles Brassard and Peter H{\o}yer and Alain Tapp},
  booktitle={Encyclopedia of Algorithms},
  year={2016}
}
In this note, 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. Assuming the function is given by a black box, this is more efficient than the best possible classical algorithm, even allowing probabilism. We also give a similar algorithm for finding claws in pairs of functions. Furthermore, we exhibit a space-time tradeoff for our technique. Our approach uses Grover’s quantum searching algorithm in a… 

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Sort L according to the second entry in each item of L

  • Sort L according to the second entry in each item of L

H;1) where H : Y ! f0; 1g denotes the function deened by H(y) = 1 if and only if a pair (x; G(y)) appears in L for some arbitrary x 2 K

  • H;1) where H : Y ! f0; 1g denotes the function deened by H(y) = 1 if and only if a pair (x; G(y)) appears in L for some arbitrary x 2 K