Constructing Polynomials for Functions over Residue Rings Modulo a Composite Number in Linear Time

  • Svetlana N. Selezneva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

We show how to check in linear time if a function \(f:{\mathbb Z}_k^n \to{\mathbb Z}_k\), where k = p m , p is a prime number, and m ≥ 2, specified by its values, can be represented by a polinomial in the ring ℤ k [x 1, …, x n ]. If so, our algorithm also constructs (in linear time) its canonical polynomial representation. We also show how to extend our techniques (with linear time) to the cases of an arbitrary composite number k.

More precisely, we prove that the circuit-size complexity of solving the problem, if a given function \(f:{\mathbb Z}_k^n \to{\mathbb Z}_k\), where k is a fixed composite number, specified by its values, is represented by a polynomial in the ring ℤ k [x 1, …, x n ] and, if so, finding its polynomial, is linear.

Keywords

Partial Order Polynomial Function Prime Number Linear Time Canonical Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Svetlana N. Selezneva
    • 1
  1. 1.Department of Computational Mathematics and CyberneticsMoscow State UniversityRussia

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