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

unpublished

We study

fatcat:xru5y6ozc5e3lktiwa5l4ki7re
*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. . ...##
###
How hard is deciding trivial versus nontrivial in the dihedral coset problem?
[article]

2016
pre-print

We study

doi:10.4230/lipics.tqc.2016.89
arXiv:1608.02003v1
fatcat:ougei5w2ovb7xagxac5ogckwmi
*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. ...##
###
Quantum Algorithms for some Hidden Shift Problems
[article]

2002
*
arXiv
*
pre-print

We also define

arXiv:quant-ph/0211140v1
fatcat:vt4kl5vvijdi5kni466g4zyche
*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 ...##
###
Quantum Algorithms for Some Hidden Shift Problems

2006
*
SIAM journal on computing (Print)
*

For one of these

doi:10.1137/s009753970343141x
fatcat:ad2sui2s7rbd3bs3r4hywbwy5e
*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 ...##
###
Relating two genus 0 problems of John Thompson
[article]

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. ...

##
###
Quantum Cryptanalysis (Dagstuhl Seminar 15371)

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*? ...

##
###
Quantum Algorithms
[article]

2008
*
arXiv
*
pre-print

amplification, quantum algorithms for simulating quantum mechanical systems, several non-

arXiv:0808.0369v1
fatcat:gsiyvpw7mnd2hlmki5tvgjwgvu
*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*. ...##
###
Two quantum Ising algorithms for the shortest-vector problem

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] . ...

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

2017
*
Lecture Notes in Computer Science
*

We establish security by treating

doi:10.1007/978-3-319-56617-7_3
fatcat:x7btopqmzbguzlsunujxqfr7be
*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*. ...##
###
On Solving Systems of Diagonal Polynomial Equations Over Finite Fields
[chapter]

2015
*
Lecture Notes in Computer Science
*

Our algorithm works

doi:10.1007/978-3-319-19647-3_12
fatcat:lx5iwkjgl5h5pmlgjhvgxnzyyu
*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*...##
###
Solving systems of diagonal polynomial equations over finite fields

2017
*
Theoretical Computer Science
*

Our algorithm works

doi:10.1016/j.tcs.2016.04.045
fatcat:54vlyohjrzaqjh3ceoeqvp3l6u
*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. ...##
###
Algorithms for group isomorphism via group extensions and cohomology
[article]

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. ...

##
###
Reflection Group Codes and Their Decoding

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. ...

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

2005
*
arXiv
*
pre-print

This included

arXiv:math/0104289v2
fatcat:ceiwm2ccmrcl3au6cit6rlm2va
*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. ...##
###
Permutation groups, minimal degrees and quantum computing
[article]

2006
*
arXiv
*
pre-print

These results contribute to

arXiv:quant-ph/0607204v1
fatcat:fayrgb2mk5esraqwtukqxs2oiq
*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*. ...
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