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First-Order Queries on Finite Abelian Groups

Simone Bova, Barnaby Martin, Marc Herbstritt
<span title="2015-09-04">2015</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/kjskj3qctjdkbigtx4fkcqiyii" style="color: black;">Annual Conference for Computer Science Logic</a> </i> &nbsp;
We prove that model checking first-order sentences on finite abelian groups is fixed-parameter tractable, when parameterized by the size of the sentence.  ...  We also prove that model checking monadic second-order sentences on finite abelian groups finitely presented by integer matrices is not fixed-parameter tractable (under standard assumptions in parameterized  ...  The authors thank Carlo Toffalori for a clarification on Baur-Monk theorem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.csl.2015.41">doi:10.4230/lipics.csl.2015.41</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/csl/BovaM15.html">dblp:conf/csl/BovaM15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e5g36yilhzbqbjrhu3jhzzdivy">fatcat:e5g36yilhzbqbjrhu3jhzzdivy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180512051829/http://drops.dagstuhl.de/opus/volltexte/2015/5406/pdf/3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d7/0f/d70f8ccd1006c9fedbee1b6e27b789d0811875f6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.csl.2015.41"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Solvable black-box group problems are low for PP

V. Arvind, N.V. Vinodchandran
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Using this we show that the problems of Membership Testing, Group Intersection, Order Verification, and Group Isomorphism over abelian black-box groups are in SPP.  ...  = {Bm}m,O be a countable family of finite groups whose elements are uniquely encoded as strings of uniform length, and group operations are computable in time bounded by a polynomial in m.  ...  We first state a fundamental theorem on finite abelian groups. Theorem 3.1 (Bumside [9] ). Let G be a finite abelian group such that JGI = p?' p? . . . p: ) where the pi's are distinct primes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0304-3975(96)00100-4">doi:10.1016/s0304-3975(96)00100-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bewtferynvatrknwc4fygjcd3q">fatcat:bewtferynvatrknwc4fygjcd3q</a> </span>
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The Hidden Subgroup Problem and Post-quantum Group-based Cryptography [article]

Kelsey Horan, Delaram Kahrobaei
<span title="2018-05-21">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in such infinite group platforms are not yet known.  ...  In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography.  ...  In general, when G is a group of finite order the HSP problem has quantum query complexity of poly(log |G|) as shown by Ettinger, Høyer and Knill [45] ; for any group G, O(log |G|) queries provides sufficient  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.04179v2">arXiv:1805.04179v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4pt43jdwqfhutiylezwk6y4w3q">fatcat:4pt43jdwqfhutiylezwk6y4w3q</a> </span>
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Control-target inversion property on Abelian groups [article]

Massoud Amini
<span title="2003-08-27">2003</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that the quantum Fourier transform on finite fields used to solve query problems is a special case of the usual quantum Fourier transform on finite abelian groups.  ...  We apply this to get a sharp query complexity separation between classical and quantum algorithms for a hidden homomorphism problem on finite Abelian groups.  ...  The quantum Fourier transform on abelian groups Let G be a finite abelian group.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0308148v1">arXiv:quant-ph/0308148v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fkcwbuifcvfsjaq7ct3vuln4du">fatcat:fkcwbuifcvfsjaq7ct3vuln4du</a> </span>
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Deterministic Algorithms for the Hidden Subgroup Problem [article]

Ashwin Nayak
<span title="2021-11-10">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely O(√( n)), where n is the order of the group.  ...  In the first version of this article (arXiv:2104.14436v1 [cs.DS]), we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup.  ...  By the fundamental theorem of abelian groups, any non-trivial finite abelian group G may be expressed as a direct product of cyclic groups of prime power order: G ∼ = Z p k 1 1 × Z p k 2 2 × • • • × Z  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2104.14436v2">arXiv:2104.14436v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gxjruvyva5ag3nhv4r4hrqmo7u">fatcat:gxjruvyva5ag3nhv4r4hrqmo7u</a> </span>
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Space Efficient Representations of Finite Groups [article]

Bireswar Das, Shivdutt Sharma, P.R.Vaidyanathan
<span title="2020-02-26">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The Cayley table representation of a group uses O(n^2) words for a group of order n and answers multiplication queries in time O(1).  ...  Farzan and Munro (ISSAC'06) defined a model for group representation and gave a succinct data structure for abelian groups with constant query-time.  ...  There is a representation of finite abelian group of order n that uses constant number of words and answers multiplication queries in constant time.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.11391v1">arXiv:2002.11391v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6trd3tkyfnfyfmeebd4ekkajzq">fatcat:6trd3tkyfnfyfmeebd4ekkajzq</a> </span>
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Quantum testers for hidden group properties [article]

Katalin Friedl, Frederic Magniez, Miklos Santha, Pranab Sen
<span title="2002-08-29">2002</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.  ...  Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers.  ...  First, we show that the tester used for LARGER-PERIOD(K) in Abelian groups yields a query efficient tester when G is any finite group and K any normal subgroup (Theorem 2).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0208184v1">arXiv:quant-ph/0208184v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/w52depprtngbhhj7quusgrfwya">fatcat:w52depprtngbhhj7quusgrfwya</a> </span>
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Quantum Testers for Hidden Group Properties [chapter]

Katalin Friedl, Frédéric Magniez, Miklos Santha, Pranab Sen
<span title="">2003</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.  ...  Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers.  ...  Acknowledgements We would like to thank Lisa Hales for giving us the basic ideas about how to extend our finite Abelian case tester to Z, and letting us include the result into this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-45138-9_36">doi:10.1007/978-3-540-45138-9_36</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/uolfzzwxefbxviuik2fkbbkrqa">fatcat:uolfzzwxefbxviuik2fkbbkrqa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170812210028/https://www.irif.fr/~santha/Papers/fmss09.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/0c/29/0c29e35d2b3a02dc9f36a91d674bcf8832a377a1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-45138-9_36"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The quantum query complexity of the abelian hidden subgroup problem

Pascal Koiran, Vincent Nesme, Natacha Portier
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
We study Simon's problem from the point of view of quantum query complexity and give here a first non-trivial lower bound on the query complexity of a hidden subgroup problem, namely Simon's problem.  ...  More generally, we give a lower bound which is optimal up to a constant factor for any abelian group.  ...  function on finite abelian groups.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2007.02.057">doi:10.1016/j.tcs.2007.02.057</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bkzyhyd4czg6tdpwyy4jmf2fwm">fatcat:bkzyhyd4czg6tdpwyy4jmf2fwm</a> </span>
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A Miniature CCA2 Public key Encryption scheme based on non-Abelian factorization problems in Lie Groups [article]

Haibo Hong, Licheng Wang, Jun Shao, Haseeb Ahmad, Yixian Yang
<span title="2016-06-10">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we first put forward a novel idea of designing cryptosystem based on Lie groups and Lie algebras.  ...  Moreover, in analog with Boyen's sceme(AsiaCrypt 2007), we disign a public key encryption scheme based on non-Abelian factorization problems in Lie Groups.  ...  [15] devised a new public key cryptographic system-M ST 3 based on random covers and LS in finite non-abelian groups.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1605.07168v3">arXiv:1605.07168v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/25qnzapxozfuhjokmceeiedycq">fatcat:25qnzapxozfuhjokmceeiedycq</a> </span>
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Abstract structure of unitary oracles for quantum algorithms

William Zeng, Jamie Vicary
<span title="2014-12-28">2014</span> <i title="Open Publishing Association"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qzezkwolyvf33pglm2jldjklca" style="color: black;">Electronic Proceedings in Theoretical Computer Science</a> </i> &nbsp;
into abelian groups.  ...  We show that a pair of complementary dagger-Frobenius algebras, equipped with a self-conjugate comonoid homomorphism onto one of the algebras, produce a nontrivial unitary morphism on the product of the  ...  Theorem 17 (Structure theorem for finite abelian groups). Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4204/eptcs.172.19">doi:10.4204/eptcs.172.19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nd6mkpompra2dpxjjq3lmgrrlm">fatcat:nd6mkpompra2dpxjjq3lmgrrlm</a> </span>
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Page 6175 of Mathematical Reviews Vol. , Issue 2000i [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In the paper under review the author classifies finite p-groups of conjugate type {1, p”} up to isoclinism, which is an equivalence relation on the class of finite groups first introduced by P.  ...  In the first part of this paper the author investigates the link between the structures of G and A when G is a finite non-abelian p-group and [G, A] is not cyclic.  ... 
<span class="external-identifiers"> </span>
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Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem [article]

Gabor Ivanyos, Frederic Magniez, Miklos Santha
<span title="2001-02-02">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Abelian normal 2-subgroup of small index or with cyclic factor group.  ...  These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary  ...  In fact, we can reduce the test to an instance of the Abelian hidden subgroup problem as follows. First, we compute the orders of the underlying elements (see [21] for example).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/quant-ph/0102014v1">arXiv:quant-ph/0102014v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2tibigzxrff4db3sp5w64kqf3u">fatcat:2tibigzxrff4db3sp5w64kqf3u</a> </span>
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Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

Gábor Ivanyos, Frédéric Magniez, Miklos Santha
<span title="">2001</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tewj77cuufbzbgbk265bb462ga" style="color: black;">Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures - SPAA &#39;01</a> </i> &nbsp;
Abelian normal 2-subgroup of small index or with cyclic factor group.  ...  These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary  ...  Permutation groups, matrix groups over finite fields and even finite matrix groups over algebraic number fields fit in this model.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/378580.378679">doi:10.1145/378580.378679</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/spaa/IvanyosMS01.html">dblp:conf/spaa/IvanyosMS01</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wgvcicdrsncbjfkunxm4enxt44">fatcat:wgvcicdrsncbjfkunxm4enxt44</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706094430/https://www.irif.fr/~magniez/PAPIERS/ims-spaa01.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/81/00/810016146a13d033167e8a6758dfdc438871f669.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/378580.378679"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

GÁBOR IVANYOS, FRÉDÉRIC MAGNIEZ, MIKLOS SANTHA
<span title="">2003</span> <i title="World Scientific Pub Co Pte Lt"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yhcgkcmjlvfovferf3in5mkmpq" style="color: black;">International Journal of Foundations of Computer Science</a> </i> &nbsp;
Abelian normal 2-subgroup of small index or with cyclic factor group.  ...  These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary  ...  Permutation groups, matrix groups over finite fields and even finite matrix groups over algebraic number fields fit in this model.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/s0129054103001996">doi:10.1142/s0129054103001996</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o5mpsjnqrfhvppj7qwpyi7j6ry">fatcat:o5mpsjnqrfhvppj7qwpyi7j6ry</a> </span>
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