A Public Key Cryptosystem Based on Non-abelian Finite Groups

Lempken, Wolfgang; Tran, Trung van; Magliveras, Spyros S.; Wen, Wei

doi:10.1007/s00145-008-9033-y

Cited by 57 publications

(64 citation statements)

References 8 publications

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“…In Section IV, we shall discuss further cryptosystems based on group factorizations [17] - [29]. Magliveras [17] in 1986 presented a secret-key cryptosystem using factorization of groups.…”

Section: ±J)mentioning

confidence: 99%

“…The first public-key system due to Qu and Vanstone [19] turned out to be vulnerable to cryptanalytic attacks using lattice reduction algorithms, [20] and [21]. New approaches to designing public-key cryptosystems using finite groups have been discussed later on [22] - [29]. Here, the group factorizations are usually denoted as logarithmic signatures.…”

Section: ±J)mentioning

confidence: 99%

“…In this direction, the authors of [29] suggested a third cryptosystem MST3 also based on group covers. Covers can be generated at random [28], allowing greater flexibility in the setup.…”

Section: Cryptosystems Based On Group Factorizationmentioning

confidence: 99%

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Group Factorizations and Information Theory

Tamm¹

2007

2007 Information Theory and Applications Workshop

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Abstract-A factorization of a group G is a collection of subsets (A 1 , A 2 , . . . , A r ) such that every element g ∈ G has a unique representation g = a 1 · a 2 · . . . · a r where a i ∈ A i for i = 1, . . . , r. We shall survey several applications of group factorizations in information theory. They occur in the analysis of syndromes of integer codes, several graphs with large girth important for LDPC codes can be constructed using group factorizations, and various cryptosystems are based on them.

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“…In Section IV, we shall discuss further cryptosystems based on group factorizations [17] - [29]. Magliveras [17] in 1986 presented a secret-key cryptosystem using factorization of groups.…”

Section: ±J)mentioning

confidence: 99%

Section: ±J)mentioning

confidence: 99%

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Group Factorizations and Information Theory

Tamm¹

2007

2007 Information Theory and Applications Workshop

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“…Logarithmic signatures were first defined by Magliveras in [18] where cryptosystem PGM was proposed using LSs. In [20], the algebraic properties of logarithmic signatures and of cryptosystem PGM were discussed by Magliveras [21], and Magliveras, Lempken, Tran van Trung and Wei proposed cryptosystem M ST 3 using logarithmic signatures and covers [17,19]. For some interesting papers studying attacks on M ST 1 , M ST 2 and M ST 3 , see [4,9,22].…”

mentioning

confidence: 98%

Minimal logarithmic signatures for finite groups of Lie type

Singhi¹,

Singhi²,

Magliveras³

2010

Des. Codes Cryptogr.

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A logarithmic signature (LS) for a finite group G is an ordered tuple α = [A 1 , A 2 , . . . , A n ] of subsets A i of G, such that every element g ∈ G can be expressed uniquely as a product g = a 1 a 2 · · · a n , where a i ∈ A i . The length of an LS α is defined to be l(α) = n i=1 |A i |. It can be easily seen that for a group G of order k j=1 p j m j , the length of any LS α for G, satisfies, l(α) ≥ k j=1 m j p j . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002. The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families P SL n (q) and P Sp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.

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“…Actually, several attempts using non-abelian algebraic structures were made and some available cryptographic schemes such as P GM , M ST 1 , M ST 2 and M ST 3 [3,16,14,11,9,24] were developed during the past decades. In particular, as a natural analogy of the hardness assumption of IFP, the group factorization problem (GFP) [10,15] and its hardness assumption over certain factorization basis, referred as logarithmic signature, play a core role in the security arguments for the family of M ST cryptosystems.…”

Section: Introductionmentioning

confidence: 99%

Minimal logarithmic signatures for one type of classical groups

Hong

Wang

Ahmad

et al. 2016

AAECC

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As a special type of factorization of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like M ST 1 , M ST 2 and M ST 3 . An LS with the shortest length, called a minimal logarithmic signature (MLS), is even desirable for cryptographic applications. The MLS conjecture states that every finite simple group has an MLS. Recently, the conjecture has been shown to be true for general linear groups GL n (q), special linear groups SL n (q), and symplectic groups Sp n (q) with q a power of primes and for orthogonal groups O n (q) with q as a power of 2. In this paper, we present new constructions of minimal logarithmic signatures for the orthogonal group O n (q) and SO n (q) with q as a power of odd primes. Furthermore, we give constructions of MLSs for a type of classical groups -projective commutator subgroup P Ω n (q).

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A Public Key Cryptosystem Based on Non-abelian Finite Groups

Cited by 57 publications

References 8 publications

Group Factorizations and Information Theory

Group Factorizations and Information Theory

Minimal logarithmic signatures for finite groups of Lie type

Minimal logarithmic signatures for one type of classical groups

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