An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

Gonçalves, Demerson Nunes; Portugal, Renato; Fernandes, Tharso D.

doi:10.5540/03.2015.003.02.0045

Cited by 2 publications

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“…When G is (one of some groups that are) a semidirect product of abelian groups, alternative efficient algorithms have been proposed. The polycyclic HSP has been addressed for Z p k ⋊ Z 2 for fixed prime power p k [40], Z q ⋉ Z p with q|(p − 1) and q = p polylog(p) , certain affine groups [41], Z m p r ⋊ Z p [42], with p ∈ P, Z p r ⋊ Z q s where p r /q = poly(log p 2 where p, q ∈ P and r, s ∈ N [43], and Z N ⋊ Zq s where N has a special prime factorization [44].…”

Section: Resultsmentioning

confidence: 99%

The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

Horan

Kahrobaei

2018

Lecture Notes in Computer Science

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In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in the proposed infinite group platforms are not yet known.

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Section: Resultsmentioning

confidence: 99%

The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

Horan

Kahrobaei

2018

Lecture Notes in Computer Science

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“…That is, given an Abelian group G and a function f : G → X being said to hide a subgroup H ≤ G if f is bijective on G/H, the goal is to find H. Jozsa [4] provided a uniform description of several important quantum algorithms such as Deutsch-Jozsa [5], Simon [2], and Shor [3] algorithms in terms of the hidden subgroup problem. Indeed, this problem has received a lot of attention where quantum algorithms were proposed for its different variants, e.g., [6,7,8,9,10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

Query complexity of generalized Simon's problem

Ye,

Huang,

et al. 2019

Preprint

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Simon's problem played an important role in the history of quantum algorithms, as it inspired Shor to discover the celebrated quantum algorithm solving integer factorization in polynomial time. Besides, the quantum algorithm for Simon's problem has been recently applied to break symmetric cryptosystems. Generalized Simon's problem is a natural extension of Simon's problem: Given a function f : {0, 1} n → {0, 1} m with n ≤ m and the promise that there exists a subgroup S ≤ Z n 2 of rank k s.t. for any s, x ∈ {0, 1} n , f (x) = f (x ⊕ s) iff s ∈ S, the goal is to find S. It is not difficult to design a quantum algorithm for solving this problem exactly with query complexity of O(n − k). However, so far it is not clear what is the classical deterministic query complexity of this problem.In this paper, we first prove that any classical deterministic algorithm solving generalized Simon's problem has to query at least Ω( √ k • 2 n−k ) values, clarifying the gap between quantum and classical computing on this problem. On the other hand, we devise a deterministic algorithm with query complexity of O( √ k • 2 n−k ) in most cases. Therefore, the obtained bound Θ( √ k • 2 n−k ) is almost optimal, which fills the blank of classical deterministic query complexity for generalized Simon's problem.

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An Efficient Quantum Algorithm for the Hidden Subgroup Problem over some Non-Abelian Groups

Cited by 2 publications

References 9 publications

The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

The Hidden Subgroup Problem and Post-quantum Group-Based Cryptography

Query complexity of generalized Simon's problem

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