This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an up-to-date bibliography of the problem.
A Class of Polynomials over Finite Fields
โ Scribed by Arnaldo Garcia; Henning Stichtenoth
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 129 KB
- Volume
- 5
- Category
- Article
- ISSN
- 1071-5797
No coin nor oath required. For personal study only.
โฆ Synopsis
Generalizing the norm and trace mappings for % O P /% O , we introduce an interesting class of polynomials over "nite "elds and study their properties. These polynomials are then used to construct curves over "nite "elds with many rational points.
๐ SIMILAR VOLUMES
We exhibit a deterministic algorithm for factoring polynomials in one variable over "nite "elds. It is e$cient only if a positive integer k is known for which I (p) is built up from small prime factors; here I denotes the kth cyclotomic polynomial, and p is the characteristic of the "eld. In the cas
Let F be a finite field. We apply a result of Thierry Berger (1996, Designs Codes Cryptography, 7, 215-221) to determine the structure of all groups of permutations on F generated by the permutations induced by the linear polynomials and any power map which induces a permutation on F.
Let T n (x, a) สฆ GF(q)[x] be a Dickson polynomial over the finite field GF(q) of either the first kind or the second kind of degree n in the indeterminate x and with parameter a. We give a complete description of the factorization of T n (x, a) over GF(q).
We study value sets of polynomials over a finite field, and value sets associated to pairs of such polynomials. For example, we show that the value sets (counting multiplicities) of two polynomials of degree at most d are identical or have at most q!(q!1)/d values in common where q is the number of