Polynomials over Finite Fields and Applications

Theoretical Computer Science, 2000

We analyse an extension of Shoup's (Inform. Process. Lett. 33 (1990) 261-267) deterministic algorithm for factoring polynomials over ÿnite prime ÿelds to arbitrary ÿnite ÿelds. In particular, we prove the existence of a deterministic algorithm which completely factors all monic polynomials of degree n over F q; q odd, except possibly O(n 2 log 2 q=q) polynomials, using O(n 2+ log 2 q) arithmetical operations in Fq.

Theoretical Computer Science, 1997

In this paper we present a new deterministic algorithm for computing the square-free decomposition of multivariate polynomials with coefficients from a finite field. Our algorithm is based on Yun's square-free factorization algorithm for characteristic 0. The new algorithm is more efficient than existing, deterministic algorithms based on Musser's squarefree algorithm. We will show that the modular approach presented by Yun has no significant performance advantage over our algorithm. The new algorithm is also simpler to implement and it can rely on any existing GCD algorithm without having to worry about choosing "good" evaluation points. To demonstrate this, we present some timings using implementations in Maple (Char et al., 1991), where the new algorithm is used for Release 4 onwards, and Axiom (Jenks and Sutor, 1992) which is the only system known to the author to use an implementation of Yun's modular algorithm mentioned above.

Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86, 1986

2011

Any non constant polynomial over a field can be expressed as a product of ir-reducible polynomials. In finite fields, some algorithms work for the calculation of irreducible factors of a polynomial of positive degree. The factorization of polynomials over finite fields has great ...

Journal of Pure and Applied Algebra, 2019

Let F q be the finite field with q elements, where q is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over F q of high degree using rational transformations. In particular, given a divisor D > 2 of q + 1 and an irreducible polynomial f ∈ F q [x] of degree n such that n is even or D ≡ 2 (mod 4), we show how to obtain from f a sequence {f i } i≥0 of irreducible polynomials over F q with deg(f i) = n • D i .

Int. J. Math. Math. Sci., 2021

For a Gaussian prime π and a nonzero Gaussian integer β � a + bi ∈ Z[i] with a≥ 1 and |β|≥ 2 + � 2 √ , it was proved that if π � αnβ n + αn− 1β n− 1 + · · · + α1β + α0≕f(β)where n≥ 1, αn ∈ Z[i]\ 0 { }, α0, . . . , αn− 1 belong to a complete residue systemmodulo β, and the digits αn− 1 and αn satisfy certain restrictions, then the polynomial f(x) is irreducible in Z[i][x]. For any quadratic field K ≔ Q( �� m √ ), it is well known that there are explicit representations for a complete residue system in K, but those of the case m ≡ 1 (mod4) are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

International Journal of Computer Applications, 2015

Irreducible Polynomials over GF(p m) and the multiplicative inverses under it are important in cryptography. Presently the method of deriving irreducible polynomials of a particular prime modulus is very primitive and time consuming. In this paper, 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. The maximum number of terms in each column of the matrix is 2 j where j is the column index.

Discrete Mathematics, 2019

Let F q be the finite field with q elements, where q is a prime power and n be a positive integer. In this paper, we explore the factorization of f (x n) over F q , where f (x) is an irreducible polynomial over F q. Our main results provide generalizations of recent works on the factorization of binomials x n − 1. As an application, we provide an explicit formula for the number of irreducible factors of f (x n) under some generic conditions on f and n.

Discrete Mathematics, 1999

2021

This paper presents a new method to factorize semi-primes using simple polynomials.We consider a semi-prime, whose factors are both congruent as represented by:According to Fermat’s Christmas Theorem, a sum of two squares can be found for each prime and two sums of two squares for the semi-prime . Using this property, we propose a new method to find the first of these sums of two squares and once this is known, the Brahmagupta identity is used to find the second sum of two squares. Subsequently, a modified Euler factorization is applied to recover the two prime constructs of the semi-prime. The correctness of our new factorisation method is established with mathematical proofs.