A New Family of Perfect Nonlinear Binomials

We prove that the binomials xp s+1 − αxpk+p2k+s define perfect nonlinear mappings inGF (p3k) for appropriate choices of the integer s and α ∈ GF (p3k). We show that these binomials are inequivalent to known perfect nonlinear monomials. As a consequence we obtain new commutative semifields for p ≥ 5 and odd k.

[1]  P. Dembowski,et al.  Planes of ordern with collineation groups of ordern2 , 1968 .

[2]  Cunsheng Ding,et al.  Signal Sets From Functions With Optimum Nonlinearity , 2007, IEEE Transactions on Communications.

[3]  Alexander Pott,et al.  A new APN function which is not equivalent to a power mapping , 2005, IEEE Transactions on Information Theory.

[4]  Eimear Byrne,et al.  An Infinite Family of Quadratic Quadrinomial APN Functions , 2007 .

[5]  Claude Carlet,et al.  A class of quadratic APN binomials inequivalent to power functions , 2006, IACR Cryptol. ePrint Arch..

[6]  Claude Carlet,et al.  New classes of almost bent and almost perfect nonlinear polynomials , 2006, IEEE Transactions on Information Theory.

[7]  Marie Henderson,et al.  Planar polynomials for commutative semifields with specified nuclei , 2007, Des. Codes Cryptogr..

[8]  Alexander Pott,et al.  Some Theorems on Planar Mappings , 2008, WAIFI.

[9]  Qing Xiang,et al.  Pseudo-Paley graphs and skew Hadamard difference sets from presemifields , 2007, Des. Codes Cryptogr..

[10]  Cunsheng Ding,et al.  A family of optimal constant-composition codes , 2005, IEEE Transactions on Information Theory.

[11]  Eimear Byrne,et al.  New families of quadratic almost perfect nonlinear trinomials and multinomials , 2008, Finite Fields Their Appl..

[12]  Tor Helleseth,et al.  Some Power Mappings with Low Differential Uniformity , 1997, Applicable Algebra in Engineering, Communication and Computing.

[13]  Kaisa Nyberg,et al.  Differentially Uniform Mappings for Cryptography , 1994, EUROCRYPT.

[14]  Robert S. Coulter,et al.  Commutative presemifields and semifields , 2008 .

[15]  C. Ding,et al.  Note A family of skew Hadamard difference sets , 2006 .

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

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

[18]  Jürgen Bierbrauer.  A family of crooked functions , 2009, Des. Codes Cryptogr..