Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields

  title={Improving the Time Complexity of the Computation of Irreducible and Primitive Polynomials in Finite Fields},
  author={Josep Rif{\`a} and Joan Borrell},
In this paper, we present a method to compute all the irreducible and primitive polynomials of degree m over a finite field. We also describe two concrete implementations of our method with respective time complexities O(m2 + m log m) and O(m2 + log m). These implementations, using in parallel different devices introduced to operate in these fields [1], [7], allows us to reduce the time complexity of our method below that of the best previously known methods [3]. Our algorithm is especially… 

A fast algorithm to compute irreducible and primitive polynomials in finite fields

This paper finds each new irreducible or primitive polynomial with a complexity ofO(m) arithmetic operations inGF(q) by using the Berlekamp-Massey algorithm.

Fast arithmetic in unramified p-adic fields

On Computing the Resultant of Generic Bivariate Polynomials

An algorithm is presented for computing the resultant of two generic bivariate polynomials over a field K using (n2 - 1/ømega d) 1+o(1) arithmetic operations in K, where two n x n matrices are multiplied using O(nømega) operations.

Normal Bases over Finite Fields

The principal result in the thesis is the complete determination of all optimal normal bases in finite fields, which confirms a conjecture by Mullin, Onyszchuk, Vanstone and Wilson.

Fast computation of special resultants

Cryptographic Counters and Applications to Electronic Voting

This work formalizes the notion of a cryptographic counter, which allows a group of participants to increment and decrement a cryptographic representation of a (hidden) numerical value privately and robustly, and shows a general and efficient reduction from any encryption scheme to a general cryptographic counter.

Fast Computation With Two Algebraic Numbers

We propose fast algorithms for computing composed multiplications and composed sums, as well as «diamond products» of univariate polynomials.

Fast construction of irreducible polynomials over finite fields

  • V. Shoup
  • Computer Science, Mathematics
    SODA '93
  • 1993
The main result of this paper a new algorithm for constructing an irreducible polynomial of specified degree n over a finite field Fq . The algorithm is probabilistic, and is asymptotically faster



Introduction to finite fields and their applications: List of Symbols

An introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications, especially information theory, algebraic coding theory and cryptology and a chapter on applications within mathematics, such as finite geometries.

Systolic Multipliers for Finite Fields GF(2m)

Two systolic architectures are developed for performing the product–sum computation AB + C in the finite field GF(2m) of 2melements, where A, B, and C are arbitrary elements of GF(2m). The first

Note for computing the minimum polynomial of elements in large finite fields

Two methods for computing the minimun polynomial of an element in the finite field GF(qm) have pratically no storage constraint and may improve by a factor 2.6 the classical method.

Efficient bit-serial multiplication and the discrete-time Wiener-Hopf equation over finite fields

It is shown that solving the DTWHE is equivalent to performing division over finite fields, and the proof provides a new interpretation of the relationship between bit- serial multiplication and DTWHEs that enables bit-serial multiplication over GF(2/sup m/) to be understood more easily.

Bit-serial Reed - Solomon encoders

  • E. Berlekamp
  • Computer Science
    IEEE Transactions on Information Theory
  • 1982
New concepts and techniques for implementing encoders for Reed-Solomon codes, with or without interleaving are presented, including only fields of order 2”, where m m ight be any integer.

Shift-register synthesis and BCH decoding

  • J. Massey
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1969
It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback

Designing Efficient Algorithms for Parallel Computers

This is it, the designing efficient algorithms for parallel computers that will be your best choice for better reading book that will not spend wasted by reading this website.

VLSI Architectures for Computing Multiplications and Inverses in GF(2m)

The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.