Construction of irreducible polynomials over finite fields

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text]. Furthermore, we construct the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text] by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

An Effective Algorithm for the Synthesis of Irreducible Polynomials over a Galois Fields of Arbitrary Characteristics

The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary characteristics. The number of testing steps coincides with the degree of the synthesized polynomials. Upon completion of testing, the polynomial is classifieds as either irreducible or composite. If the degree of the synthesized polynomials is small (no more than two dozen), the formation of a set of tested polynomials is carried out using the exhaustive search method. For large values of the degree, the test polynomials are generating by statistical modeling. The developed algorithm allows one to synthesize binary irreducible polynomials up to 2Kbit on personal computers of average performance

Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].