New algorithms for finding irreducible polynomials over finite fields

@article{Shoup1988NewAF,
  title={New algorithms for finding irreducible polynomials over finite fields},
  author={Victor Shoup},
  journal={[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science},
  year={1988},
  pages={283-290}
}
  • V. Shoup
  • Published 24 October 1988
  • Mathematics, Computer Science
  • [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
An algorithm is presented for finding an irreducible polynomial of specified degree over a finite field. It is deterministic and runs in polynomial time for fields of small characteristics. A proof is given of the stronger result, that the problem of finding irreducible polynomials of specified degree over a finite field K is deterministic-polynomial-time reducible to the problem of factoring polynomials over the prime field of K.<<ETX>> 

Factorization of Polynomials over the Field of Rational Numbers

In this paper, a randomized algorithm for deciding the irreducibility of an irreducible polynomial and factoring a reducible polynomial over the field of rational numbers is presented. The main idea

An algorithm for determining the irreducible polynomials over finite fields

We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square

Construction of Irreducible Polynomials

This chapter is devoted to the problem of constructing irreducible polynomials over a given finite field. Such polynomials are used to implement arithmetic in extension fields and are found in many

A correspondence of certain irreducible polynomials over finite fields

An approach to the problem of generating irreducible polynomials over the finite field GF(2) and its relationship with the problem of periodicity on the space of binary sequences

A method for generating irreducible polynomials of degree n over the finite field GF(2) is proposed. The irreducible polynomials are found by solving a system of equations that brings the information

Systematic Generation of An Irreducible Polynomial of an Arbitrary Degree m over Fp Such That p ≫ m

TLDR
The proposed method for generating an irreducible polynomial of an arbitrary degree m over an arbitrary prime field Fp such that p > m has the following features: its complexity has little dependency on the size of characteristic p, its calculation cost is explicitly given with degree m, and it can generate primitive polynomials when pm - 1 is factorized as the product of prime numbers.

Finding irreducible and primitive polynomials

  • I. Shparlinski
  • Mathematics
    Applicable Algebra in Engineering, Communication and Computing
  • 2005
TLDR
It is shown, that for anyQ large enough one can design a finite field F with q withq=Q + o(Q) elements in polynomial time (log Q)o(1).

An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

TLDR
In order to find all irreducible polynomials, be it monic or non-monic, of all prime moduli p with all its order m, a fast deterministic computer algorithm based on an algebraic method producing a (m×m) matrix is proposed.

Fast arithmetic for the algebraic closure of finite fields

TLDR
This work presents algorithms to construct and do arithmetic operations in the algebraic closure of the finite field Fp, inspired by algorithms for constructing irreducible polynomials, to give efficient algorithms for embeddings and isomorphisms.

Factoring Polynomials over Special Finite Fields

TLDR
This work exhibits a deterministic algorithm for factoring polynomials in one variable over finite fields that requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which ?k(p) has a prime factor l with l?1 mod r.
...

References

SHOWING 1-10 OF 44 REFERENCES

Factorization of multivariate polynomials over finite fields

We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of

Irreducible Polynomials over Finite Fields

  • J. Gathen
  • Mathematics, Computer Science
    FSTTCS
  • 1986
TLDR
Several methods of computing irreducible polynomials over finite fields are presented and the estimates for the preprocessing time depend on unproven conjectures.

Probabilistic Algorithms in Finite Fields

  • M. Rabin
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1980
We present probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynomial, and factoring a polynomial into its

Factoring polynomials over large finite fields*

TLDR
Some of the known algorithms for factoring polynomials over finite fields are reviewed and a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) is presented.

Finding irreducible polynomials over finite fields

TLDR
Irreducible polynomials in Fp[X] are used to carry out the arithmetic in field extension of Fp to solve the random polynomial time problem of finding irreducibles of any degree over Fp.

Factoring Polynomials and Primitive Elements for Special Primes

  • J. Gathen
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1987

On the Deterministic Complexity of Factoring Polynomials over Finite Fields

  • V. Shoup
  • Computer Science, Mathematics
    Inf. Process. Lett.
  • 1990

Equations over Finite Fields: An Elementary Approach

Equations yd=f(x) and yq?y=f(x).- Character sums and exponential sums.- Absolutely irreducible equations f(x,y)=0.- Equations in many variables.- Absolutely irreducible equations f(x1,...,xn)=0.-

Very Fast Parallel Matrix and Polynomial Arithmetic

  • W. Eberly
  • Mathematics, Computer Science
    FOCS
  • 1984
We present very efficient arithmetic circuits for the computation of the determinants and inverse of band matrices and for interpolation and polynomial arithmetic over arbitrary ground fields. For

Equations over Finite Fields

We have seen that for each prime p, there is a field F p of p elements. In fact, given any prime p and an integer r ≥ 1, there is one and only one field F q of q = p r elements. The field F q ⊇ F p