A permutation polynomial over a finite field 
is a polynomial ![f in F_q[X]](/images/equations/PermutationPolynomial/Inline2.svg)
for which 
is a permutation
of 
.
For example, 
induces the identity permutation on 
. An o-polynomial
is a normalized permutation polynomial satisfying additional permutation conditions
(Chen and Huang 2026).
Permutation Polynomial
See also
Finite Field, o-Polynomial, PermutationExplore with Wolfram|Alpha
References
Chen, B. and Huang, J. "Deep Holes of Full-Length Reed-Solomon Codes." 21 Oct 2026. https://arxiv.org/abs/2610.29429.Lidl, R. and Niederreiter, H. Finite Fields, 2nd ed. Cambridge, England: Cambridge University Press, 1996. https://doi.org/10.1017/CBO9780511525926.Cite this as:
Weisstein, Eric W. "Permutation Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PermutationPolynomial.html