A linear decomposition attack

Myasnikov, Alexei; Roman'kov, V. A.

doi:10.1515/gcc-2015-0007

Cited by 34 publications

(30 citation statements)

References 45 publications

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“…As we have already pointed out, if a (semi)group G is non-commutative and has non-central invertible elements, then it always has a non-identical inner automorphism, i.e., conjugation by an element g ∈ G such that g −1 hg = h for at least some h ∈ G. Now let G be the semigroup of 3 × 3 matrices over the group ring Z 7 [A 5 ], where A 5 is the alternating group on 5 elements. Here we use an extension of the semigroup G by an inner automorphism ϕ H , which is conjugation by a matrix As we have mentioned in the Introduction, the protocol in this section was attacked in [9] and [14] by a "linear algebra attack". This was possible partly because of the special "compact" form of the above security assumptions, and partly because the dimension of a linear representation of the platform semigroup happens to be small enough in this case for a linear algebra attack to be computationally feasible.…”

Section: Matrices Over Group Rings and Extensions By Inner Automorphismsmentioning

confidence: 99%

“…We therefore offer here another platform group that we believe should make the protocol invulnerable to the attacks of [3], [9], [14]. The group is a free nilpotent p-group, for a sufficiently large prime p. We give a formal definition of this group in Section 8; here we just say that this is a finite group all of whose elements have order dividing p n for some fixed n ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

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Using Semidirect Product of (Semi)groups in Public Key Cryptography

Kahrobaei

Shpilrain

2016

Pursuit of the Universal

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Abstract. In this survey, we describe a general key exchange protocol based on semidirect product of (semi)groups (more specifically, on extensions of (semi)groups by automorphisms), and then focus on practical instances of this general idea. This protocol can be based on any group or semigroup, in particular on any non-commutative group. One of its special cases is the standard Diffie-Hellman protocol, which is based on a cyclic group. However, when this protocol is used with a noncommutative (semi)group, it acquires several useful features that make it compare favorably to the Diffie-Hellman protocol. The focus then shifts to selecting an optimal platform (semi)group, in terms of security and efficiency. We show, in particular, that one can get a variety of new security assumptions by varying an automorphism used for a (semi)group extension.

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Section: Matrices Over Group Rings and Extensions By Inner Automorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Using Semidirect Product of (Semi)groups in Public Key Cryptography

Kahrobaei

Shpilrain

2016

Pursuit of the Universal

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“…In [22], the author introduced the method of linear decomposition applicable in algebraic cryptanalysis. In [19] this method was further developed by the author and Myasnikov. See also [21], [25] and [24].…”

Section: Introductionmentioning

confidence: 99%

An improvement of the Diffie–Hellman noncommutative protocol

Roman'kov

2021

Des. Codes Cryptogr.

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The main purpose of this paper is to propose a new version of the Diffie-Hellman noncommutative key exchange protocol invented in 2000 by Ko, Lee, Cheon, Han, Kang, and Park. This new version is resistant to linear algebra attacks. It is based on a new complex algorithmic problem using the concept of a marginal set. In particular, it is resistant to attacks by the methods of Cheon and Jun and Tsaban, as well as to attacks by the methods of linear and nonlinear decompositions, developed by the author.

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“…Such a protocol developed in a spirit of Noncommutative Cryptography (NC), see [17]- [24]). It is very important that Non-Commutative cryptography is well supported by new modern achievements in Cryptanalysis (see [40] - [48]).…”

Section: Introductionmentioning

confidence: 99%

On the usage of postquantum protocols defined in terms of transformation semi- groups and their homomophisms

Ustimenko¹

2020

TACS

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We suggest new applications of protocols of Non-commutative cryptography defined in terms of subsemigroups of Affine Cremona Semigroups over finite commutative rings and their homomorphic images to the constructions of possible instruments of Post Quantum Cryptography. This approach allows to define cryptosystems which are not public keys. When extended protocol is finished correspondents have the collision multivariate transformation on affine space or variety ( * ) where is a finite commutative ring and * is nontrivial multiplicative subgroup of . The security of such protocol rests on the complexity of word problem to decompose element of Affine Cremona Semigroup given in its standard form into composition of given generators. The collision map can serve for the safe delivery of several bijective multivariate maps (generators) on from one correspondent to another. So asymmetric cryptosystem with nonpublic multivariate generators where one side (Alice) knows inverses of but other does not have such a knowledge is possible. We consider the usage of single protocol or combi-nations of two protocols with platforms of different nature. The usage of two protocols with the collision spaces and ( * ) allows safe delivery of two sets of generators of different nature. In terms of such sets we define an asymmetric encryption scheme with the plainspace ( * ) , cipherspace and multivariate non-bijective encryption map of unbounded degree ( ) and polynomial density on with injective restriction on ( * ) . Algebraic cryptanalysis faces the problem to interpolate a natural decryption transformation which is not a map of polynomial density.

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A linear decomposition attack

Cited by 34 publications

References 45 publications

Using Semidirect Product of (Semi)groups in Public Key Cryptography

Using Semidirect Product of (Semi)groups in Public Key Cryptography

An improvement of the Diffie–Hellman noncommutative protocol

On the usage of postquantum protocols defined in terms of transformation semi- groups and their homomophisms

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