New binomial bent functions over the finite fields of odd characteristic

The p-ary binomial function f(x) mapping GF(p4k) to GF(p) given by f(x) = Tr4k equations is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems. Finally, we characterize the size of a cyclotomic coset containing the exponent of a monomial bent functions (the same result holds in a binomial case) and provide numerical data.

[1]  Tor Helleseth,et al.  Some results about the cross-correlation function between two maximal linear sequences , 1976, Discret. Math..

[2]  Jacques Wolfmann,et al.  The weights of the dual code of the MELAS code over GF(3) , 1989, Discret. Math..

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

[4]  C. Carlet,et al.  On generalized bent and q-ary perfect nonlinear functions , 1999, Proceedings of the 1999 IEEE Information Theory and Communications Workshop (Cat. No. 99EX253).

[5]  Tor Helleseth,et al.  A note on the cross-correlation function between two binary maximal length linear sequences , 1978, Discret. Math..

[6]  John J. Komo,et al.  Nonbinary Kasami sequences over GF(p) , 1992, IEEE Trans. Inf. Theory.

[7]  P. Vijay Kumar,et al.  Generalized Bent Functions and Their Properties , 1985, J. Comb. Theory, Ser. A.

[8]  Tor Helleseth,et al.  New Families of Almost Perfect Nonlinear Power Mappings , 1999, IEEE Trans. Inf. Theory.

[9]  Yoji Niho.  Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences , 1972 .

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

[11]  Tor Helleseth,et al.  On P-Ary Bent Functions Defined on Finite Fields , 2003 .

[12]  Yin Tan,et al.  Association schemes arising from bent functions , 2011, Des. Codes Cryptogr..

[13]  Yin Tan,et al.  Strongly regular graphs associated with ternary bent functions , 2010, J. Comb. Theory, Ser. A.

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

[15]  Oscar Moreno,et al.  Prime-phase sequences with periodic correlation properties better than binary sequences , 1991, IEEE Trans. Inf. Theory.

[16]  Xiang-Dong Hou,et al.  p-Ary and q-ary versions of certain results about bent functions and resilient functions , 2004, Finite Fields Their Appl..

[17]  Robert S. Coulter,et al.  Planar Functions and Planes of Lenz-Barlotti Class II , 1997, Des. Codes Cryptogr..

[18]  G. Lachaud,et al.  The weights of the orthogonals of the extended quadratic binary Goppa codes , 1990, IEEE Trans. Inf. Theory.