Generalized Maiorana-McFarland class and normality of p-ary bent functions

Abstract A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.

[1]  Tor Helleseth,et al.  New binomial bent functions over the finite fields of odd characteristic , 2010, IEEE Trans. Inf. Theory.

[2]  Wilfried Meidl,et al.  On the dual of (non)-weakly regular bent functions and self-dual bent functions , 2013, Adv. Math. Commun..

[3]  Pascale Charpin,et al.  Normal Boolean functions , 2004, J. Complex..

[4]  Gregor Leander,et al.  Construction of bent functions from near-bent functions , 2009, J. Comb. Theory, Ser. A.

[5]  Keqin Feng,et al.  Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications , 2007, IEEE Transactions on Information Theory.

[6]  Wilfried Meidl,et al.  Bent Functions of Maximal Degree , 2012, IEEE Transactions on Information Theory.

[7]  Tor Helleseth,et al.  New binomial bent functions over the finite fields of odd characteristic , 2009, 2010 IEEE International Symposium on Information Theory.

[8]  Wilfried Meidl,et al.  A construction of bent functions from plateaued functions , 2013, Des. Codes Cryptogr..

[9]  Tor Helleseth,et al.  Generalized bent functions and their relation to Maiorana-McFarland class , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[10]  Jing Yang,et al.  A recursive construction of p-ary bent functions which are not weakly regular , 2010, 2010 IEEE International Conference on Information Theory and Information Security.

[11]  Rudolf Lide,et al.  Finite fields , 1983 .

[12]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[13]  W. Meidl,et al.  Not) weakly regular univariate bent functions , 2013, 1310.8071.

[14]  Tor Helleseth,et al.  Monomial and quadratic bent functions over the finite fields of odd characteristic , 2006, IEEE Transactions on Information Theory.

[15]  Claude Carlet,et al.  Normal extensions of bent functions , 2004, IEEE Transactions on Information Theory.

[16]  Hans Dobbertin,et al.  Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity , 1994, FSE.

[17]  Tor Helleseth,et al.  A Class of Binomial Bent Functions Over the Finite Fields of Odd Characteristic , 2012, IEEE Transactions on Information Theory.

[18]  Anne Canteaut,et al.  Finding nonnormal bent functions , 2006, Discret. Appl. Math..

[19]  Pascale Charpin,et al.  On bent and semi-bent quadratic Boolean functions , 2005, IEEE Transactions on Information Theory.

[20]  Wilfried Meidl,et al.  A construction of weakly and non-weakly regular bent functions , 2010, J. Comb. Theory, Ser. A.

[21]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.