Quantum algorithms for some hidden shift problems
Wim van Dam,Sean Hallgren,Lawrence Ip +2 more
- pp 489-498
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TLDR
In this article, the authors present three examples of unknown shift problems that can be solved efficiently on a quantum computer using the quantum Fourier transform and define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.Citations
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A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem
TL;DR: A quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$.
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Constructing elliptic curve isogenies in quantum subexponential time
TL;DR: This work gives a new subexponential-time quantum algorithm for constructing nonzero isogenies between two such elliptic curves, assuming the Generalized Riemann Hypothesis (but with no other assumptions).
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Constructing elliptic curve isogenies in quantum subexponential time
TL;DR: In this article, a quantum algorithm for constructing an isogeny between two elliptic curves is presented, where the isogenies from an elliptic curve E to itself form the endomorphism ring of the curve; this ring is an imaginary quadratic order O∆ of discriminant ∆ < 0.
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Hidden translation and orbit coset in quantum computing
TL;DR: Efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series are given.
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Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
TL;DR: This work gives polynomial-time quantum algorithms for three problems from computational algebraic number theory, including Pell's equation, the principal ideal problem in real quadratic number fields, and the one-way function underlying the Buchmann--Williams key exchange system.
References
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Book
Quantum Computation and Quantum Information
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Book
Handbook of Applied Cryptography
TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.
Journal ArticleDOI
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
TL;DR: In this paper, the authors considered factoring integers and finding discrete logarithms on a quantum computer and gave an efficient randomized algorithm for these two problems, which takes a number of steps polynomial in the input size of the integer to be factored.
Journal ArticleDOI
Quantum Complexity Theory
Ethan Bernstein,Umesh Vazirani +1 more
TL;DR: This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.