A note on permutation polynomials and finite geometries

Abstract A polynomial ƒ over a finite field F is called a difference permutation polynomial if the mapping x → ƒ( x + a ) − ƒ( x ) is a permutation of F for each nonzero element a of F . Difference permutation polynomials give rise to affine planes. We show that when F = GF( p ), where p is a prime, the only difference permutation polynomials over F are quadratic.