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How Hard Is Deciding Trivial Versus Nontrivial in the Dihedral Coset Problem?

Nai-Hui Chia, Sean Hallgren
unpublished
We study the hardness of the dihedral hidden subgroup problem.  ...  We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density > 1 but approaching 1 as the problem size increases  ...  We first define the dihedral coset space problem and show how to use it to solve the dihedral coset problem. Then we define the coset space which we wish to understand. .  ... 
fatcat:xru5y6ozc5e3lktiwa5l4ki7re

How hard is deciding trivial versus nontrivial in the dihedral coset problem? [article]

Nai-Hui Chia, Sean Hallgren
2016 pre-print
We study the hardness of the dihedral hidden subgroup problem.  ...  We also show that if a unitary can exactly decide membership in the coset subspace, then the collision problem for subset sum can be solved for density >1 but approaching 1 as the problem size increases  ...  We first define the dihedral coset space problem and show how to use it to solve the dihedral coset problem. Then we define the coset space which we wish to understand.  ... 
doi:10.4230/lipics.tqc.2016.89 arXiv:1608.02003v1 fatcat:ougei5w2ovb7xagxac5ogckwmi

Quantum Algorithms for some Hidden Shift Problems [article]

Wim van Dam, Lawrence Ip
2002 arXiv   pre-print
We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.  ...  In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform.  ...  Acknowledgments We would like to thank the anonymous referee who pointed out the application of shifted Legendre symbol problem to algebraically homomorphic cryptosystems and Umesh Vazirani, whose many  ... 
arXiv:quant-ph/0211140v1 fatcat:vt4kl5vvijdi5kni466g4zyche

Quantum Algorithms for Some Hidden Shift Problems

Wim van Dam, Sean Hallgren, Lawrence Ip
2006 SIAM journal on computing (Print)  
For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems  ...  We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem.  ...  We would like to thank the anonymous referee who pointed out the application of the shifted Legendre symbol problem to algebraically homomorphic cryptosystems, and Umesh Vazirani, whose many suggestions  ... 
doi:10.1137/s009753970343141x fatcat:ad2sui2s7rbd3bs3r4hywbwy5e

Relating two genus 0 problems of John Thompson [article]

Michael D. Fried
2009 arXiv   pre-print
The "relating" entwines three problems: 1. Davenport's Problem, describing pairs of polynomials over Q whose ranges on Z/p are the same for almost all p. 2.  ...  representations, and dihedral and cyclic groups. 3.  ...  It seems somewhere in the literature is the phrase genus 0 problem attached to a specific Hecke formulation.  ... 
arXiv:0910.3974v1 fatcat:ifadhre3ajfgbjuntrh4knibvu

Quantum Cryptanalysis (Dagstuhl Seminar 15371)

Michele Mosca, Martin Roetteler, Nicolas Sendrier, Rainer Steinwandt, Marc Herbstritt
2016 Dagstuhl Reports  
In this seminar, participants explored the impact that quantum algorithms will have on cryptology once a large-scale quantum computer becomes available.  ...  This report documents the program and the outcomes of Dagstuhl Seminar 15371 "Quantum Cryptanalysis".  ...  How Hard is Deciding Trivial versus Nontrivial in the Dihedral Coset Problem?  ... 
doi:10.4230/dagrep.5.9.1 dblp:journals/dagstuhl-reports/MoscaRSS15 fatcat:a3bpfckt3fespm27nxmaqykx6i

Quantum Algorithms [article]

Michele Mosca
2008 arXiv   pre-print
amplification, quantum algorithms for simulating quantum mechanical systems, several non-trivial generalizations of the Abelian Hidden Subgroup Problem (and related techniques), the quantum walk paradigm  ...  It is infeasible to detail all the known quantum algorithms, so a representative sample is given.  ...  Classical algorithms for the Abelian Hidden Subgroup Problem In the black-box model Ω( |G/K|) queries are necessary in order to even decide if the hidden subgroup is trivial.  ... 
arXiv:0808.0369v1 fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu

Two quantum Ising algorithms for the shortest-vector problem

David Joseph, Adam Callison, Cong Ling, Florian Mintert
2021 Physical Review A  
In this paper we describe two variants of a quantum Ising algorithm to solve this problem.  ...  New cryptosystems are being designed and standardized for the postquantum era, and a significant proportion of these rely on the hardness of problems like the shortest-vector problem to a quantum adversary  ...  The dihedral coset problem is another type of HSP; a relaxed form, the extrapolated dihedral coset problem, has been shown to be equivalent to LWE [26] .  ... 
doi:10.1103/physreva.103.032433 fatcat:snckx27nyvbc5h2t7h7to5wkxu

Quantum-Secure Symmetric-Key Cryptography Based on Hidden Shifts [chapter]

Gorjan Alagic, Alexander Russell
2017 Lecture Notes in Computer Science  
We establish security by treating the (quantum) hardness of the well-studied Hidden Shift problem as a basic cryptographic assumption.  ...  We observe that this problem has a number of attractive features in this cryptographic context, including random self-reducibility, hardness amplification, and--in many cases of interest--a reduction from  ...  There is also evidence connecting HSP on the dihedral group D N (and hence also HS on Z/N ) to other hard problems.  ... 
doi:10.1007/978-3-319-56617-7_3 fatcat:x7btopqmzbguzlsunujxqfr7be

On Solving Systems of Diagonal Polynomial Equations Over Finite Fields [chapter]

Gábor Ivanyos, Miklos Santha
2015 Lecture Notes in Computer Science  
Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant.  ...  class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant.  ...  Indeed, this problem has a remarkable worst case versus average case hardness property: solving it on the average is at least as hard as solving various lattice problems in the worst case, such as the  ... 
doi:10.1007/978-3-319-19647-3_12 fatcat:lx5iwkjgl5h5pmlgjhvgxnzyyu

Solving systems of diagonal polynomial equations over finite fields

Gábor Ivanyos, Miklos Santha
2017 Theoretical Computer Science  
Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant.  ...  class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant.  ...  The research is partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant "Random numbers from quantum processes," MOE2012-T3-1-009.  ... 
doi:10.1016/j.tcs.2016.04.045 fatcat:54vlyohjrzaqjh3ceoeqvp3l6u

Algorithms for group isomorphism via group extensions and cohomology [article]

Joshua A. Grochow, Youming Qiao
2017 arXiv   pre-print
The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in n^ n+O(1) time, but only recently were polynomial-time algorithms designed for several interesting group  ...  The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions  ...  Q. was supported by the Australian Research Council DECRA DE150100720. Both authors were supported by NSF grant DMS-1620484 during the preparation of the manuscript.  ... 
arXiv:1309.1776v2 fatcat:te4qq4kzm5acvimwjrjqbi3mxi

Reflection Group Codes and Their Decoding

W. Wesley Peterson, J. B. Nation, Marc P. Fossorier
2010 IEEE Transactions on Information Theory  
The complexity of decoding is analyzed, and it is shown that a proper choice of the sequence of subgroups used in the algorithm can yield significant gains in the efficiency of decoding.  ...  The new algorithm is proved to achieve maximum likelihood decoding.  ...  ACKNOWLEDGMENT The authors would like to thank the referees for some useful suggestions.  ... 
doi:10.1109/tit.2010.2080571 fatcat:36eeekyesnhnzdhaozyvf5ntyq

Hurwitz monodromy, spin separation and higher levels of a modular tower [article]

Paul Bailey, Michael D. Fried
2005 arXiv   pre-print
This included the regular version of the Inverse Galois Problem.  ...  This includes systematic exposure of moduli spaces having points where the field of moduli is a field of definition and other points where it is not.  ...  The history of the Inverse Galois Problem shows it is hard to find Q realizations of most finite groups G.  ... 
arXiv:math/0104289v2 fatcat:ceiwm2ccmrcl3au6cit6rlm2va

Permutation groups, minimal degrees and quantum computing [article]

Julia Kempe, Laszlo Pyber, Aner Shalev
2006 arXiv   pre-print
These results contribute to the foundations of a non-commutative coding theory. A main application of our results concerns the Hidden Subgroup Problem for the symmetric group in Quantum Computing.  ...  We completely characterize the hidden subgroups of the symmetric group that can be distinguished from identity with weak Quantum Fourier Sampling, showing these are exactly the subgroups with bounded minimal  ...  In this case it is not hard to see [HRT00, GSVV01] that the probability to sample ρ is independent of the coset of H we happen to land in.  ... 
arXiv:quant-ph/0607204v1 fatcat:fayrgb2mk5esraqwtukqxs2oiq
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