A Linear Algebra Attack on the Non-commuting Cryptography Class Based on Matrix Power Function

Liu, Jinhui; Zhang, Huanguo; Jia, Jianwei

doi:10.1007/978-3-319-54705-3_21

Cited by 10 publications

(20 citation statements)

References 14 publications

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“…Since the methods of solution of MPF NSR S are unknown yet and there is not even any theory on how to handle such a kind of system of equations, we hypothesize that the proposed key length in practice will provide sufficient security against a total scan attack and the attack presented in [27].…”

Section: Discussionmentioning

confidence: 99%

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Sakalauskas

2018

Symmetry

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Abstract:A new enhanced matrix power function (MPF) is presented for the construction of cryptographic primitives. According to the definition in previously published papers, an MPF is an action of two matrices powering some base matrix on the left and right. The MPF inversion equations, corresponding to the MPF problem, are derived and have some structural similarity with classical multivariate quadratic (MQ) problem equations. Unlike the MQ problem, the MPF problem seems to be more complicated, since its equations are not defined over the field, but are represented as left-right action of two matrices defined over the infinite near-semiring on the matrix defined over the certain infinite, additive, noncommuting semigroup. The main results are the following: (1) the proposition of infinite, nonsymmetric, and noncommuting algebraic structures for the construction of the enhanced MPF, satisfying associativity conditions, which are necessary for cryptographic applications; (2) the proof that MPF inversion is polynomially equivalent to the solution of a certain kind of generalized multivariate quadratic (MQ) problem which can be reckoned as hard; (3) the estimation of the effectiveness of direct MPF value computation; and (4) the presentation of preliminary security analysis, the determination of the security parameter, and specification of its secure value. These results allow us to make a conjecture that enhanced MPF can be a candidate one-way function (OWF), since the effective (polynomial-time) inversion algorithm for it is not yet known. An example of the application of the proposed MPF for the Key Agreement Protocol (KAP) is presented. Since the direct MPF value is computed effectively, the proposed MPF is suitable for the realization of cryptographic protocols in devices with restricted computation resources.

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Section: Discussionmentioning

confidence: 99%

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Sakalauskas

2018

Symmetry

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“…However, it is shown in Liu et al (2016), that this asymmetric key exchange is vulnerable to a certain linear algebra attack. Furthermore, their idea also holds in case of asymmetric encryption proposed in Mihalkovich and Sakalauskas (2012).…”

Section: Our Previous Workmentioning

confidence: 99%

“…In their paper (Liu et al, 2016) have also suggested an idea of using non-commuting (semi)group to define a platform structure, i.e. the entries of base matrix Q should not commute.…”

Section: The Analysis Linear Algebra Attackmentioning

confidence: 99%

“…Furthermore, in Mihalkovich and Sakalauskas (2012) we presented an asymmetric encryption protocol, which unfortunately is not resistant to linear algebra attack described in Liu et al (2016). The key-point of this attack is eliminating matrix U by replacing it with its polynomial expression.…”

Section: The Analysis Linear Algebra Attackmentioning

confidence: 99%

“…where E is a matrix representing the function value. In this paper we mainly focus on papers (Sakalauskas et al, 2008;Mihalkovich and Sakalauskas, 2012;Sakalauskas et al, 2017) and (Liu et al, 2016). In the latter paper authors present an attack based on linear algebra, which can be applied to protocols, presented in Sakalauskas et al (2008) and Mihalkovich and Sakalauskas (2012) to break them in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

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Improved Asymmetric Cipher Based on Matrix Power Function Resistant to Linear Algebra Attack

Sakalauskas¹,

Mihalkovich²

2017

Informatica

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In this paper we consider an improved version of earlier published asymmetric encryption protocol based on matrix power function (MPF). Recently, a linear algebra attack on earlier version of this protocol was found. This attack allows an attacker to break suggested protocol in polynomial time. Here we show that the improved version of our encryption protocol is not vulnerable to the declared linear attack, while retaining its effective realization in embedded systems.

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