Discrete logarithm like problems and linear recurring sequences

Santos González; Llorenç Huguet; Consuelo Martínez; Hugo Villafañe

doi:10.3934/amc.2013.7.187

Article Contents

2013, Volume 7, Issue 2: 187-195. Doi: 10.3934/amc.2013.7.187

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Discrete logarithm like problems and linear recurring sequences

1.
Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo
2.
Departament de Ciènces Matemàtiques i Informàtica, Universitat de les Illes Balears, Ctra. de Valldemossa km 7, 5, 07122 Palma de Mallorca

Received: September 30, 2012

Published: April 2013

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Abstract

In this paper we study the hardness of some discrete logarithm like problems defined in linear recurring sequences over finite fields from a point of view as general as possible. The intractability of these problems plays a key role in the security of the class of public key cryptographic constructions based on linear recurring sequences. We define new discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems for any nontrivial linear recurring sequence in any finite field whose minimal polynomial is irreducible. Then, we prove that these problems are polynomially equivalent to the discrete logarithm, Diffie-Hellman and decisional Diffie-Hellman problems in the subgroup generated by any root of the minimal polynomial of the sequence.

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Mathematics Subject Classification: Primary: 11T71; Secondary: 94A55.

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