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Efficient Inversion of Rational Maps Over Finite Fields

  • Antonio Cafure
  • Guillermo Matera
  • Ariel Waissbein
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

We study the problem of finding the inverse image of a point in the image of a rational map F : \( \mathbb{F}_q^n \to \mathbb{F}_q^n \) over a finite field \( \mathbb{F}_q \). Our interest mainly stems from the case where F encodes a permutation given by some public-key cryptographic scheme. Given an element y (0)F(\( \mathbb{F}_q^n \)), we are able to compute the set of values x (0)\( \mathbb{F}_q^n \) for which F(x (0)= y (0) holds with O(Tn 4.38 D 2.38δlog2 q) bit operations, up to logarithmic terms. Here T is the cost of the evaluation of F 1,..., F n, D is the degree of F and δ is the degree of the graph of F.

Key words

Finite fields polynomial system solving public-key cryptography matrices of fixed displacement rank 

AMS(MOS) subject classifications

14G05 68W30 11T71 8Q25 47B35 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Antonio Cafure
    • 3
    • 4
  • Guillermo Matera
    • 1
    • 4
  • Ariel Waissbein
    • 2
    • 5
  1. 1.CONICETArgentina
  2. 2.ITBACdad. de Buenos AiresArgentina
  3. 3.Depto. de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  4. 4.Instituto del Desarrollo HumanoUniversidad Nacional de General SarmientoLos PolvorinesArgentina
  5. 5.CoreLabsCore Security TechnologiesCiudad de Buenos AiresArgentina

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