Power functions with low uniformity on odd characteristic finite fields

In this paper, we give some new low differential uniformity of some power functions defined on finite fields with odd characteristic. As corollaries of the uniformity, we obtain two families of almost perfect nonlinear functions in GF(3n) and GF(5n) separately. Our results can be used to prove the Dobbertin et al.’s conjecture.

[1]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

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

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

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

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

[6]  Hans Dobbertin,et al.  One-to-One Highly Nonlinear Power Functions on GF(2n) , 1998, Applicable Algebra in Engineering, Communication and Computing.

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

[8]  Hans Dobbertin,et al.  Almost Perfect Nonlinear Power Functions on GF(2n): The Niho Case , 1999, Inf. Comput..

[9]  Hans Dobbertin,et al.  Almost Perfect Nonlinear Power Functions on GF(2n): The Welch Case , 1999, IEEE Trans. Inf. Theory.

[10]  H. Dobbertin.  Almost Perfect Nonlinear Power Functions on GF(2n): A New Case for n Divisible by 5 , 2001 .

[11]  Hans Dobbertin,et al.  APN functions in odd characteristic , 2003, Discret. Math..

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

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

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

[15]  C. Ding,et al.  Explicit classes of permutation polynomials of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % , 2009 .

[16]  Cunsheng Ding,et al.  Explicit classes of permutation polynomials of $$ \mathbb{F}_{3^{3m} } $$ , 2009 .