On multiple output bent functions

In this article we investigate the possibilities of obtaining multiple output bent functions from certain power polynomials over finite fields. So far multiple output bent functions F:GF(2)^n->GF(2)^m (where n is even and m=GF(2)^m, is a multiple output bent function. We consider all the main cases of monomial trace bent functions and specify the restrictions on @l and m that yield multiple output bent functions F(x)=Tr"m^n(@lx^d). Interestingly enough, in one particular case when n=4r, d=(2^r+1)^2, a multiple bent function F(x)=Tr"2"r^n(ax^d) could not be obtained by considering a collection of 2r Boolean bent functions of the form f"i(x)=Tr"1^n(@l"ix^d) for some suitable coefficients @l"i@?GF(2^n).

[1]  Guang Gong,et al.  Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar , 2005 .

[2]  Claude Carlet,et al.  Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems , 1998, Des. Codes Cryptogr..

[3]  Amr M. Youssef,et al.  Hyper-bent Functions , 2001, EUROCRYPT.

[4]  Kaisa Nyberg,et al.  Perfect Nonlinear S-Boxes , 1991, EUROCRYPT.

[5]  Anne Canteaut,et al.  Construction of bent functions via Niho power functions , 2006, J. Comb. Theory, Ser. A.

[6]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

[7]  Jing Yang,et al.  Vectorial Boolean Functions with Good Cryptographic Properties , 2011, Int. J. Found. Comput. Sci..

[8]  Gregor Leander,et al.  Monomial bent functions , 2006, IEEE Transactions on Information Theory.

[9]  Mitsuru Matsui,et al.  Linear Cryptanalysis Method for DES Cipher , 1994, EUROCRYPT.

[10]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..

[11]  Guang Gong,et al.  Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials , 2008, IEEE Trans. Inf. Theory.

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

[13]  Wang Xuan-ming,et al.  An Optimized Method for Multiple Output Bent Functions , 2005 .

[14]  Guang Gong,et al.  Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials , 2008, IEEE Transactions on Information Theory.

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

[16]  Claude Carlet,et al.  Two New Classes of Bent Functions , 1994, EUROCRYPT.

[17]  Anne Canteaut,et al.  A new class of monomial bent functions , 2006, 2006 IEEE International Symposium on Information Theory.

[18]  Pascale Charpin,et al.  Cubic Monomial Bent Functions: A Subclass of M , 2008, SIAM J. Discret. Math..