Polynomial Automorphisms and the Jacobian Conjecture

Essen, Arno van den; Kuroda, Shigeru; Crachiola, Anthony J.

doi:10.1007/978-3-030-60535-3

Cited by 205 publications

(105 citation statements)

References 28 publications

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“…Note that (Φ( 1 )( ), … , Φ( )( )) = (Φ( 1 )( ), … , Φ( )( )), and this implies that the compounds are equal, so Φ( )( ) = Φ( )( ) for each , but the map Φ is injective so ( ) = ( ) for each ∈ {1, … , } this shows that = for each ∈ {1, … , } which implies that = which means that must be injective. Using the results from 10 , it can be seen that is invertible, and this completes our proof.…”

Section: Discussionsupporting

confidence: 62%

Attacking Jacobian Problem Using Resultant Theory

Jony¹,

Al-Rashed²

2022

Baghdad Sci.J

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Section: Discussionsupporting

confidence: 62%

Attacking Jacobian Problem Using Resultant Theory

Jony¹,

Al-Rashed²

2022

Baghdad Sci.J

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“…An invertibility criterion for endomorphisms of polynomial algebras can be obtained in terms of symbolic computation and Gröbner bases. For a comprehensive reference we refer to the book [25].…”

Section: Difference Stream Ciphersmentioning

confidence: 99%

An algebraic attack to the Bluetooth stream cipher E0

Scala¹,

Polese²,

Tiwari³

et al. 2022

Preprint

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In this paper we study the security of the Bluetooth stream cipher E0 from the viewpoint it is a difference stream cipher, that is, it is defined by a system of explicit difference equations over the finite field GF(2). This approach highlights some issues of the Bluetooth encryption as the invertibility of its state transition map, a special set of 14 bits of its 132-bit state which when guessed imply linear equations among the other bits and finally a very small number of spurious keys compatible with a keystream of about 60 bits. Exploiting such issues, we implement an algebraic attack using Gröbner bases, SAT solvers and Binary Decision Diagrams. Testing activities suggest that the version based on Gröbner bases is the best one and it is able to attack E0 in about 2 79 seconds on an Intel i9 CPU. To the best of our knowledge, this work improves any previous attack based on a short keystream, hence fitting with Bluetooth specifications.

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“…In particular, it is known that if a polynomial map P : C n → C n is injective, then it is surjective and so is a polynomial automorphism. For more details on this open problem, we refer the reader to the survey paper [26].…”

Section: Analytic Automorphisms Of Subspaces Of C Nmentioning

confidence: 99%

Analytic Automorphisms and Transitivity of Analytic Mappings

Novosad

Zagorodnyuk

2020

Mathematics

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In this paper, we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms that show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator that is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclicity of composition operators.

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Polynomial Automorphisms and the Jacobian Conjecture

Cited by 205 publications

References 28 publications

Attacking Jacobian Problem Using Resultant Theory

Attacking Jacobian Problem Using Resultant Theory

An algebraic attack to the Bluetooth stream cipher E0

Analytic Automorphisms and Transitivity of Analytic Mappings

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