On the cycle structure of permutation polynomials

Any permutation of a finite field Fq can be represented by a polynomial Pn(x)=(⋯+((a0x+a1)q−2+a2)q−2+⋯+an)q−2+an+1, for some n⩾0. P0 is linear and the cycle structure of P1 is known. In this work we present the cycle structure of the polynomials P2(x) and P3(x) completely and give methods for constructing Pn(x) with full cycle, for arbitrary n⩾1.