Cubic permutation polynomials and elliptic curves

Abstract

In this thesis, we study the elliptic curve E : y² = f(x), where f(x) is a cubic permutation polynomial over some finite commutative ring R. In case R is the finite field F[subscript q], it turns out that the group of rational points on E is cyclic of order q+1. This group is a product of cyclic groups if R = Z[subscript n] or Z[i]/(α), the ring of integers modulo a square-free $n$ and the ring of Guassian integers modulo a square-free $\alpha$, respectively. In addition, we introduce a shift-invariant elliptic curve which is an elliptic curve E : y² = f(x), where y² - f(x) is a weak permutation polynomial. We give a necessary and sufficient condition for the existence of a shift-invariant elliptic curve over F[subscript q], Z[subscript n] and Z[i] / (α).