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Efficient Inversion of Rational Maps Over Finite Fields

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Algorithms in Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 146))

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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.

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Author information

Authors and Affiliations

  1. CONICET, Argentina

    Guillermo Matera

  2. ITBA, Av. Eduardo Madero 399, (C1106ACD), Cdad. de Buenos Aires, Argentina

    Ariel Waissbein (Doctorado en Ingeniería)

  3. Depto. de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, (C1428EHA), Buenos Aires, Argentina

    Antonio Cafure

  4. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, (1613), Los Polvorines, Argentina

    Antonio Cafure & Guillermo Matera

  5. CoreLabs, Core Security Technologies, Humboldt 1967, (C1414CTU), Ciudad de Buenos Aires, Argentina

    Ariel Waissbein (Doctorado en Ingeniería)

Authors
  1. Antonio Cafure
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  2. Guillermo Matera
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  3. Ariel Waissbein
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Editor information

Editors and Affiliations

  1. Departamento de Matemática Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428), Buenos Aires, Argentina

    Alicia Dickenstein

  2. Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123, Saarbrücken, Germany

    Frank-Olaf Schreyer

  3. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556-4618, USA

    Andrew J. Sommese

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Cafure, A., Matera, G., Waissbein, A. (2008). Efficient Inversion of Rational Maps Over Finite Fields. In: Dickenstein, A., Schreyer, FO., Sommese, A.J. (eds) Algorithms in Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 146. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75155-9_4

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