(I may be the last person on Earth to know this, but...) All "even" perm polys over a prime p can be written in the form: p(x) = (a + (b + c + ... + (z+x)^(p-2))...)^(p-2)) e.g., p1(x) = a+x p2(x) = a+(b+x)^(p-2) p3(x) = a+(b+(c+x)^(p-2))^(p-2) p4(x) = a+(b+(c+(d+x)^(p-2))^(p-2))^(p-2) etc. So, for p=7, we have p1(x) = a+x p2(x) = a+(b+x)^5 p3(x) = a+(b+(c+x)^5))^5 p4(x) = a+(b+(c+(d+x)^5)^5)^5 p5(x) = a+(b+(c+(d+(e+x)^5)^5)^5)^5 p6(x) = a+(b+(c+(d+(e+(f+x)^5)^5)^5)^5 This form may not look familiar, but we can rewrite y^5 as 1/y, and we now recognize this form as a *continued fraction* !! Thus, for p6(x), a,b,c,d,e,f forms a continued fraction for the permutation polynomial !! Note that this cf representation is not unique, but it may be possible to produce a canonic form. We can get *odd* permutations by premultiplying the entire permutation polynomial by a constant from the other coset.