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Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
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Cited by 174 (13 self)
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We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
A framework for fast quantum mechanical algorithms
"... A framework is presented for the design and analysis of quantum mechanical algorithms, the O ( N) step quantum search algorithm is an immediate consequence of this framework. It leads to several other searchtype applications an example is presented where the WalshHadamard (WH) transform of the q ..."
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Cited by 97 (1 self)
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A framework is presented for the design and analysis of quantum mechanical algorithms, the O ( N) step quantum search algorithm is an immediate consequence of this framework. It leads to several other searchtype applications an example is presented where the WalshHadamard (WH) transform of the quantum search algorithm is replaced by another transform tailored to the parameters of the problem. Also, it leads to quantum mechanical algorithms for problems not immediately connected with search two such algorithms are presented for calculating the mean and median of statistical distributions. In order to classically estimate either the mean or median of a given distribution to a precision ε, needs Ω ε 2 – steps. The best known quantum mechanical algorithm for estimating the median takes steps, and that for estimating the mean takes O ε 1 –
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 27 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 24 (2 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
Random Oracles in a Quantum World
"... Abstract. The interest in postquantum cryptography — classical systems that remain secure in the presence of a quantum adversary — has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have ..."
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Cited by 17 (3 self)
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Abstract. The interest in postquantum cryptography — classical systems that remain secure in the presence of a quantum adversary — has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have classical access to the random oracle. We argue that to prove postquantum security one needs to prove security in the quantumaccessible random oracle model where the adversary can query the random oracle with quantum state. We begin by separating the classical and quantumaccessible random oracle models by presenting a scheme that is secure when the adversary is given classical access to the random oracle, but is insecure when the adversary can make quantum oracle queries. We then set out to develop generic conditions under which a classical random oracle proof implies security in the quantumaccessible random oracle model. We introduce the concept of a historyfree reduction which is a category of classical random oracle reductions that basically determine oracle answers independently of the history of previous queries, and we prove that such reductions imply security in the quantum model. We then show that certain postquantum proposals, including ones based on lattices, can be proven secure using historyfree reductions and are therefore postquantum secure. We conclude with a rich set of open problems in this area.
On the security and the efficiency of the Merkle signature scheme
, 2005
"... This paper builds on the multitime signature scheme proposed by Merkle. We prove that the original scheme is existentially unforgeable under adaptive chosen message attack. Moreover, we present an improved version which has three advantages: It is provably forward secure. The number of signatures t ..."
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Cited by 9 (0 self)
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This paper builds on the multitime signature scheme proposed by Merkle. We prove that the original scheme is existentially unforgeable under adaptive chosen message attack. Moreover, we present an improved version which has three advantages: It is provably forward secure. The number of signatures that can be made with one private key is  in a practical sense  unlimited. Finally, the cost for key generation is kept low. The theoretical exposition is complemented...
A Faster Lattice Reduction Method Using Quantum Search
, 2003
"... We propose a new lattice reduction method. Our algorithm approximates shortest lattice vectors up to a factor and makes use of Grover's quantum search algorithm. The proposed method has the expected running time O(n A). That is about the square root of the running time O(n ..."
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Cited by 7 (0 self)
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We propose a new lattice reduction method. Our algorithm approximates shortest lattice vectors up to a factor and makes use of Grover's quantum search algorithm. The proposed method has the expected running time O(n A). That is about the square root of the running time O(n A) of Schnorr's recent random sampling reduction which in turn improved the running time to the fourth root of previously known algorithms. Our result demonstrates that the availability of quantum computers will a#ect not only the security of cryptosystems based on integer factorization or discrete logarithms, but also of lattice based cryptosystems. Rough estimates based on our asymptotic improvements and experiments reported in [HPS98] suggest that the NTRU security parameter needed to be increased from 503 to 1277 if su#ciently large quantum computer were available nowadays.
An improved claw finding algorithm using quantum walk
, 2007
"... The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N ≤ M), respectively, and the same range, the goal of the problem is to find x and y suc ..."
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Cited by 3 (0 self)
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The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N ≤ M), respectively, and the same range, the goal of the problem is to find x and y such that f (x)=g(y). This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm using quantum walk that solves this problem. Our algorithm can be slightly modified to solve a more general problem of finding a tuple consisting of elements in the two function domains that has prespecified property. Our algorithm can also be generalized to find a claw of k functions for any constant integer k>1, where the domains of the functions may have different size. Keywords: quantum computing, query complexity, oracle computation 1