A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem

@article{Kuperberg2005ASQ,
  title={A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem},
  author={Greg Kuperberg},
  journal={SIAM J. Comput.},
  year={2005},
  volume={35},
  pages={170-188}
}
  • G. Kuperberg
  • Published 14 February 2003
  • Computer Science
  • SIAM J. Comput.
We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity $2^{O(\sqrt{\log\ N})}$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast, the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character… 

Figures and Tables from this paper

On Quantum Sieve Approaches to the Lattice Shortest Vector Problem

time, and Regev’s reduction gives a quadratic blowup in the input size. This paper tries to give a deeper reason why Kuperberg’s algorithm does not work to provide a subexponential algorithm for

Hidden Translation and Translating Coset in Quantum Computing

TLDR
The self-reducibility framework, combined with Kuperberg's subexp exponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

Quantum algorithms for algebraic problems

TLDR
This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem

  • D. Bacon
  • Mathematics
    Quantum Inf. Comput.
  • 2008
TLDR
A new representation theoretic explanation for the pretty good measurement derived algorithm for efficiently solving the Heisenberg hidden subgroup problem is given and evidence that Clebsch-Gordan transforms over finite groups are a new primitive in quantum algorithm design is provided.

A fusion algorithm for solving the hidden shift problem in finite abelian groups

TLDR
The central tool is an extension of Peikert's adaptation of Kuperberg’s collimation sieve to arbitrary finite abelian groups that allows for a reduction to the hidden shift problem in the quotient G/2pG, which can be tackled in polynomial time.

Quantum Security of the Legendre PRF

TLDR
This paper gives two algorithms that recover the key of a shifted Legendre symbol with unknown shift, with a complexity smaller than exhaustive search of the key, a quantum variant of the table-based collision algorithm and Kuperberg’s abelian hidden shift algorithm in an offline manner.

A new quantum algorithm for the hidden shift problem in $\mathbb{Z}_{2^t}^n$

TLDR
A solution to the case when k is a power of 2, which has polynomial running time in n, and only uses quadratic classical, and linear quantum space in n log(k).

Quantum algorithms for typical hard problems: a perspective of cryptanalysis

TLDR
This paper discussed the designing methodology, algorithm framework and latest progress of the mathematic hard problems on which the typical cryptosystems depend, including integer factorization problem, discrete logarithmic problem and its variants, lattice problem, dihedral hidden subgroup problems and extrapolated dihedral coset problem.

SeaSign: Compact isogeny signatures from class group actions

TLDR
A new signature scheme for isogenies is given that combines the class group actions of CSIDH with the notion of Fiat-Shamir with aborts, and is potentially shorter than lattice signatures, but signing and verification are currently very expensive.

Hidden Shift Quantum Cryptanalysis and Implications

TLDR
The starting point of the paper is a follow up of these previous results: replace the common bitwise addition with other operations, as a modular addition, to counter Simon's quantum attack.
...

References

SHOWING 1-10 OF 29 REFERENCES

Groups with Representations of Bounded Degree

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that

Noise-tolerant learning, the parity problem, and the statistical query model

TLDR
The paper describes a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise and shows that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries.

Quantum computation of Fourier transforms over symmetric groups

TLDR
A quantum polynomial time algorithm for the Fourier transform for the symmetric groups Sn is given, adapting results obtained by Clausen and Diaconis–Rockmore to the quantum setting.

20世紀の名著名論:Peter Shor : Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

Shorの論文は,そのアルゴリズム開発でのひらめき, 理論的解析の深さと美しさだけでも名論文に値するもの だが,最も重要な点は 21世紀の直前に新しいコンピュー タモデルとしての量子力学原理を情報処理に用いた量子 コンピュータの研究を先導したということだ.色々な観 点はあるかもしれないが,この論文が 21世紀で量子計 算と呼ばれている分野を爆発的に広め,いま少し歴史の

Linear representations of finite groups

Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero:

Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on

Optimal On-Line Simulations of Tree Machines by Random Access Machines

This paper shows that every tree machine of time complexity t can be simulated on-line by a log-cost random access machine (RAM) of time complexity $O((t\log t)/\log \log t)$. Using

Quantum computation and quantum information

  • T. Paul
  • Physics
    Mathematical Structures in Computer Science
  • 2007
This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal

On Time Versus Space

TLDR
The context-sensitive languages cannot be recognized in linear time by deterministic multitape Turing machines, and are strictly contained in the class of languages recognized by Turing machines of tape complexity.

Generic quantum Fourier transforms

TLDR
This paper uses Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families.