Abstract
Finding new classes of permutation polynomials is a challenging problem. Blockhuis at al. investigated the permutation behavior of polynomials of the form \(\sum_{i=0}^{n-1}a_iX^{2^i+1}\) over \({\mathbb F}_{{2^n}}\). In this paper, we extend their results and propose as a new conjecture that if n = 2e then X 2 is the only unitary permutation polynomial of this type.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Lidl, R., Mullen, G.: When does a Polynomial over a Finite Field Permute the Elements of the Field? Amer. Math. Monthly 100, 71–74 (1993)
Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge University Press, Cambridge (1997)
Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)
Dobbertin, H.: Almost Perfect Nonlinear Power Functions on GF(2N): The Niho Case. Inf. Comput. 151(1-2), 57–72 (1999)
Budaghyan, L., Carlet, C., Leander, G.: A Class of Quadratic APN Binomials Inequivalent to Power Functions. Cryptology ePrint Archive, Report 2006/445 (2006), http://eprint.iacr.org/
Blokhuis, A., Coulter, R.S., Henderson, M., O’Keefe, C.M.: Permutations Amongst the Dembowski-Ostrom Polynomials. In: 1999 Finite Fields and Applications, pp. 37–42. Springer, Berlin (2001)
Payne, S.: A Complete Determination of Translation Ovoids in Finite Desarguian Planes. Lincei - Rend. Sc. fis. mat. e nat. (1971)
Berger, T., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: Almost Perfect Nonlinear Functions. Technical Report RR-5774, INRIA Rocquencourt (2005), http://www.inria.fr/rrrt/rr-5774.html
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Laigle-Chapuy, Y. (2007). A Note on a Class of Quadratic Permutations over \({\mathbb F}_{{2^n}}\) . In: Boztaş, S., Lu, HF.(. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2007. Lecture Notes in Computer Science, vol 4851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77224-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-540-77224-8_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77223-1
Online ISBN: 978-3-540-77224-8
eBook Packages: Computer ScienceComputer Science (R0)