Leiden University Student Repository

Beerens, Pjotr (2024)

Families of Defining Polynomials of Finite Fields through Prime Power Torsion on Elliptic Curves

Bachelor thesis | Wiskunde (BSc)

We present a method to generate irreducible polynomials of degree p^n in F_p[X] by considering elements of order p^{n+1} on elliptic curves given by a Weierstrass equation over F_p. For this method, we provide and prove a necessary and sufficient condition on the elliptic curve in order for the generated polynomials to be irreducible. We do so by investigating the Frobenius morphism of the elliptic curve and its action on the p^k-torsion subgroups of the curve. In particular, the presented method generates irreducible polynomials if and only if the number of F_p-rational points on the elliptic curve is either p or p + 2. Finally, we provide a non-trivial lower bound on the multiplicative order of the roots of these defining polynomials in an extension field.

Author

Beerens, Pjotr

Faculty

Faculty of Science

Supervisors

Bruin, P.J.
Zidaric, N.

ECTS Credits

24

Language

en