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The problem of distinguishing between a random function and a random permutation on a domain of size $N$ is important in theoretical cryptography, where the security of many primitives depend on the problem's hardness.
How to construct random functions
Oded Goldreich, Shafi Goldwasser, and Silvio Micali · 1986
Earlier work this paper cites.
How to construct pseudorandom permutations from pseudorandom functions
Michael Luby and Charles Rackoff · 1988
Earlier work this paper cites.
Strengths and weaknesses of quantum computing
Charles H Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani · 1997
Earlier work this paper cites.
Quantum algorithm for the collision problem
Gilles Brassard, Peter Hoyer, and Alain Tapp · 1997
Earlier work this paper cites.
Quantum mechanics helps in searching for a needle in a haystack
Lov K Grover · 1997
Earlier work this paper cites.
On the power of quantum computation
Daniel R Simon · 1997
Earlier work this paper cites.
Tight bounds on quantum searching
Michel Boyer, Gilles Brassard, Peter Høyer, and Alain Tapp · 1998
Earlier work this paper cites.
On the construction of pseudorandom permutations: LubyÑrackoff revisited
Moni Naor and Omer Reingold · 1999
Cited alongside, same era.
Quantum lower bounds by quantum arguments
Andris Ambainis · 2000
Cited alongside, same era.
Quantum lower bounds by polynomials
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf · 2001
Cited alongside, same era.
Quantum lower bounds for the collision and the element distinctness problems
Scott Aaronson and Yaoyun Shi · 2004
Cited alongside, same era.
Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range
Andris Ambainis · 2005
Cited alongside, same era.
Probability and computing: Randomized algorithms and probabilistic analysis
Michael Mitzenmacher and Eli Upfal · 2005
Cited alongside, same era.
Negative weights make adversaries stronger
Peter Hoyer, Troy Lee, and Robert Spalek · 2007
Later among the works it cites.
A short proof of the prp/prf switching lemma
Donghoon Chang and Mridul Nandi · 2008
Later among the works it cites.
The need for structure in quantum speedups
Scott Aaronson and Andris Ambainis · 2011
Later among the works it cites.
How to construct quantum random functions
Mark Zhandry · 2012
Later among the works it cites.
Secure identity-based encryption in the quantum random oracle model
Mark Zhandry · 2012
Later among the works it cites.
A note on the quantum collision problem for random functions
Mark Zhandry · 2013
Closest in time.
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