A Note on a Class of Quadratic Permutations over F2n

1 Finding new classes of permutation polynomials is a chal-lenging problem with applications in cryptography, coding theory andin combinatorial designs [1]. We will here focus on a specific instanceraised by Patarin in his paper introducig HFE [2], namely characterizingquadratic permutation polynomials. After a survey of existing results, wewill present our extension of some results from Blokhuis at al. [3] dealingwith the permutation behavior of polynomials of the form Pn−1i=0 aiX 2+1overF2n . We will also propose as a new conjecture that if n = 2 thenX is the only unitary permutation polynomial of this type. References1. Lidl, R., Mullen, G.: When does a polynomial over a finite field permute the elementsof the field? Amer. Math. Monthly 100 (1993) 71–742. Patarin, J.: Hidden fields equations (HFE) and isomorphisms of polynomials (IP):Two new families of asymmetric algorithms. In: EUROCRYPT. (1996) 33–483. Blokhuis, A., Coulter, R.S., Henderson, M., O’Keefe, C.M.: Permutations amongstthe Dembowski-Ostrom polynomials. In: Finite fields and applications (Augsburg,1999), Berlin, Springer (2001) 37–42 1 This work will be presented at AAECC-17