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Four recursive constructions of permutation polynomials over $\gf(q^2)$ withthose over $\gf(q)$ are developed and applied to a few famous classes ofpermutation polynomials. They produce infinitely many new permutationpolynomials over $\gf(q^{2^\ell})$ for any positive integer $\ell$ with anygiven permutation polynomial over $\gf(q)$. A generic construction ofpermutation polynomials over $\gf(2^{2m})$ with o-polynomials over $\gf(2^m)$is also presented, and a number of new classes of permutation polynomials over$\gf(2^{2m})$ are obtained.

Five Constructions of Permutation Polynomials over $\gf(q^2)$