A conjecture on permutation trinomials over finite fields of characteristic two

In this paper, by analyzing the quadratic factors of an $11$-th degree polynomial over the finite field $\ftwon$, a conjecture on permutation trinomials over $\ftwon[x]$ proposed very recently by Deng and Zheng is settled, where $n=2m$ and $m$ is a positive integer with $\gcd(m,5)=1$.

[1]  June-Bok Lee,et al.  Permutation polynomials and group permutation polynomials , 2001, Bulletin of the Australian Mathematical Society.

[2]  Xiang-dong Hou,et al.  Determination of a type of permutation trinomials over finite fields, II , 2013, Finite Fields Their Appl..

[3]  Cunsheng Ding,et al.  Permutation polynomials of the form cx+Trql/q(xa)$cx+\text {Tr}_{q^{l}/ q}(x^{a})$ and permutation trinomials over finite fields with even characteristic , 2014, Cryptography and Communications.

[4]  Cunsheng Ding,et al.  Optimal Ternary Cyclic Codes From Monomials , 2013, IEEE Transactions on Information Theory.

[5]  Qiang Wang,et al.  Cyclotomic Mapping Permutation Polynomials over Finite Fields , 2007, SSC.

[6]  Tor Helleseth,et al.  Several classes of permutation trinomials from Niho exponents , 2016, Cryptography and Communications.

[7]  Michael E. Zieve.  On some permutation polynomials over Fq of the form x^r h(x^{(q-1)/d}) , 2007, 0707.1110.

[8]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[9]  Yann Laigle-Chapuy,et al.  Permutation polynomials and applications to coding theory , 2007, Finite Fields Their Appl..

[10]  R. K. Sharma,et al.  Some new classes of permutation trinomials over finite fields with even characteristic , 2016, Finite Fields Their Appl..

[11]  Dabin Zheng,et al.  More classes of permutation trinomials with Niho exponents , 2018, Cryptography and Communications.

[12]  Lei Hu,et al.  Further results on permutation trinomials over finite fields with even characteristic , 2017, Finite Fields Their Appl..

[13]  Tor Helleseth,et al.  New permutation quadrinomials over F22m , 2018, Finite Fields Their Appl..

[14]  Xiangyong Zeng,et al.  Two classes of permutation trinomials with Niho exponents , 2018, Finite Fields Their Appl..

[15]  Xiang-dong Hou,et al.  A class of permutation trinomials over finite fields , 2013, 1303.0568.

[16]  Kangquan Li,et al.  New Permutation Trinomials Constructed from Fractional Polynomials , 2016, ArXiv.

[17]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[18]  Cunsheng Ding,et al.  Permutation trinomials over F2m , 2017, Finite Fields Their Appl..

[19]  Tao Zhang,et al.  Some new results on permutation polynomials over finite fields , 2015, Des. Codes Cryptogr..

[20]  Xi Chen,et al.  New classes of permutation binomials and permutation trinomials over finite fields , 2015, Finite Fields Their Appl..

[21]  涂自然.  Two classes of permutation polynomials having the form (x2m+x +δ)s+x , 2015 .

[22]  Jorg Schwenk,et al.  Public key encryption and digital signatures based on permutation polynomials , 1998 .

[23]  Lei Hu,et al.  Several classes of complete permutation polynomials , 2014, Finite Fields Their Appl..

[24]  Cunsheng Ding,et al.  A family of skew Hadamard difference sets , 2006, J. Comb. Theory, Ser. A.

[25]  Xiang-dong Hou,et al.  Permutation polynomials over finite fields - A survey of recent advances , 2015, Finite Fields Their Appl..

[26]  Xiangyong Zeng,et al.  A survey on the applications of Niho exponents , 2018, Cryptography and Communications.

[27]  Tor Helleseth,et al.  A class of new permutation trinomials , 2018, Finite Fields Their Appl..

[28]  Tor Helleseth,et al.  New permutation trinomials from Niho exponents over finite fields with even characteristic , 2018, Cryptography and Communications.