On permutation polynomials of the form x^1+2k + L(x)

Abstract

We investigate the open problem of Li and Wang [On EA-equivalence of certain permutations to power mappings, Des. Codes Cryptogr. 58(2011), pp. 259–269] that whether the polynomial $x^{1+2^k}+L\left(x\right)$ on $GF\left(2^n\right)$ is a permutation for $\gcd (k,n)>1$. Several classes of polynomials of the form $x^{1+2^k}+L\left(x\right)$ are proven to be permutations by decomposing the finite field $\mathrm{GF}\left(2^n\right)$ when $n\equiv 0\left(\mathrm{mod}3\right)$. Some relationships among this type of permutation polynomials, which allow certain secondary constructions, are also proposed.

Keywords:

• permutation polynomial
• linearized polynomial
• trace function
• finite field
• Bentfunction

2000 AMS Subject Classifications::

• 06E30
• 11T06
• 94A60

Acknowledgements

The authors would like to thank Professor Xiang-Dong Hou for his help in proving the theorems in Section 3.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Xin Gong and Wenfen Liu is supported by the National Basic Research Program of China under Grants [2012CB315905, 2012CB315901], and National Nature Science Foundation of China under Grant [61379150]. The work of Guangpu Gao is supported by National Nature Science Foundation of China under Grant [61402522] and the Foundation of State Key Laboratory of Information Security under Grant [2015-MS-07].