August  2019, 13(3): 505-512. doi: 10.3934/amc.2019031

A conjecture on permutation trinomials over finite fields of characteristic two

1. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China

2. 

State Key Laboratory of Cryptology, P.O. Box 5159, Beijing 100878, China

Received  August 2018 Published  April 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (Nos.61702166, 61761166010) and the National Natural Science Foundation of Hubei Province of China (No. 2017CFB143).

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

Citation: Nian Li, Qiaoyu Hu. A conjecture on permutation trinomials over finite fields of characteristic two. Advances in Mathematics of Communications, 2019, 13 (3) : 505-512. doi: 10.3934/amc.2019031
References:
[1]

H. Deng and D. Zheng, More classes of permutation trinomials with Niho exponents, Cryptogr. Commun., 11 (2019), 227-236.  doi: 10.1007/s12095-018-0284-7.

[2]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.

[3]

C. DingL. QuQ. WangJ. Yuan and P. Yuan, Permutation trinomials over finite fields with even characteristic, SIAM Journal on Discrete Mathematics, 29 (2015), 79-92.  doi: 10.1137/140960153.

[4]

C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006.

[5]

R. Gupta and R. K. Sharma, Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 41 (2016), 89-96.  doi: 10.1016/j.ffa.2016.05.004.

[6]

X. Hou, A class of permutation trinomials over finite fields, Acta Arith., 162 (2014), 51-64.  doi: 10.4064/aa162-1-3.

[7]

X. Hou, Determination of a type of permutation trinomials over finite fields Ⅱ, Finite Fields Appl., 35 (2015), 16-35.  doi: 10.1016/j.ffa.2015.03.002.

[8]

X. Hou, Permutation polynomials over finite fields - a survey of recent advances, Finite Fields Appl., 32 (2015), 82-119.  doi: 10.1016/j.ffa.2014.10.001.

[9]

Y. Laigle-Chapuy, Permutation polynomial and applications to coding theory, Finite Fields Appl., 13 (2007), 58-70.  doi: 10.1016/j.ffa.2005.08.003.

[10]

K. LiL. Qu and X. Chen, New classes of permutation binomials and permutation trinomials over finite fields, Finite Fields Appl., 43 (2017), 69-85.  doi: 10.1016/j.ffa.2016.09.002.

[11]

K. LiL. QuC. Li and S. Fu, New permutation trinomials constructed from fractional polynomials, Acta Arith., 183 (2018), 101-116.  doi: 10.4064/aa8461-11-2017.

[12]

N. Li and T. Helleseth, Several classes of permutation trinomials from Niho exponents, Cryptogr. Commun., 9 (2017), 693-705.  doi: 10.1007/s12095-016-0210-9.

[13]

N. Li and T. Helleseth, New permutation trinomials from Niho exponents over finite fields with even characteristic, Cryptogr. Commun., 11 (2019), 129-136.  doi: 10.1007/s12095-018-0321-6.

[14]

N. Li and X. Zeng, A survey on the applications of Niho exponents, Cryptogr. Commun., 2018, 1–40. doi: 10.1007/s12095-018-0305-6.

[15]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. Cambridge University Press, 1997.

[16]

J. MaT. ZhangT. Feng and G. Ge, Some new results on permutation polynomials over finite fields, Des. Codes Cryptogr., 83 (2017), 425-443.  doi: 10.1007/s10623-016-0236-1.

[17]

Y. H. Park and J. B. Lee, Permutation polynomials and group permutation polynomials, Bull. Austral. Math. Soc., 63 (2001), 67-74.  doi: 10.1017/S0004972700019110.

[18]

R. L. RivestA. Shamir and L. M. Adelman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (1978), 120-126.  doi: 10.1145/359340.359342.

[19]

J. Schwenk and K. Huber, Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34 (1998), 759-760.  doi: 10.1049/el:19980569.

[20]

Z. Tu and X. Zeng, Two classes of permutation trinomials with Niho exponents, Finite Fields Appl., 53 (2018), 99-112.  doi: 10.1016/j.ffa.2018.05.007.

[21]

Z. TuX. Zeng and T. Helleseth, New permutation quadrinomials over $\mathbb{F}_{2^{2m}}$, Finite Fields Appl., 50 (2018), 304-318.  doi: 10.1016/j.ffa.2017.11.013.

[22]

Z. TuX. ZengC. Li and T. Helleseth, A class of new permutation trinomials, Finite Fields Appl., 50 (2018), 178-195.  doi: 10.1016/j.ffa.2017.11.009.

[23]

Z. TuX. Zeng and L. Hu, Several classes of complete permutation polynomials, Finite Fields Appl., 25 (2014), 182-193.  doi: 10.1016/j.ffa.2013.09.007.

[24]

Z. TuX. Zeng and Y. Jiang, Two classes of permutation polynomials having the form $(x^{2^m}+x+\delta)^s+x$, Finite Fields Appl., 31 (2015), 12-24.  doi: 10.1016/j.ffa.2014.09.005.

[25]

Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Lecture Notes in Comput. Sci., 4893 (2007), 119-128.  doi: 10.1007/978-3-540-77404-4_11.

[26]

D. WuP. YuanC. Ding and Y. Ma, Permutation trinomials over $\mathbb{F}_{2^m}$, Finite Fields Appl., 46 (2017), 38-56.  doi: 10.1016/j.ffa.2017.03.002.

[27]

Z. ZhaL. Hu and S. Fan, Further results on permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 45 (2017), 43-52.  doi: 10.1016/j.ffa.2016.11.011.

[28]

M. Zieve, On some permutation polynomials over $\mathbb{F}_q$ of the form $x^rh(x^{\frac{q-1}{d}})$, Proc. Amer. Math. Soc., 137 (2009), 2209-2216.  doi: 10.1090/S0002-9939-08-09767-0.

show all references

References:
[1]

H. Deng and D. Zheng, More classes of permutation trinomials with Niho exponents, Cryptogr. Commun., 11 (2019), 227-236.  doi: 10.1007/s12095-018-0284-7.

[2]

C. Ding and T. Helleseth, Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59 (2013), 5898-5904.  doi: 10.1109/TIT.2013.2260795.

[3]

C. DingL. QuQ. WangJ. Yuan and P. Yuan, Permutation trinomials over finite fields with even characteristic, SIAM Journal on Discrete Mathematics, 29 (2015), 79-92.  doi: 10.1137/140960153.

[4]

C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006.

[5]

R. Gupta and R. K. Sharma, Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 41 (2016), 89-96.  doi: 10.1016/j.ffa.2016.05.004.

[6]

X. Hou, A class of permutation trinomials over finite fields, Acta Arith., 162 (2014), 51-64.  doi: 10.4064/aa162-1-3.

[7]

X. Hou, Determination of a type of permutation trinomials over finite fields Ⅱ, Finite Fields Appl., 35 (2015), 16-35.  doi: 10.1016/j.ffa.2015.03.002.

[8]

X. Hou, Permutation polynomials over finite fields - a survey of recent advances, Finite Fields Appl., 32 (2015), 82-119.  doi: 10.1016/j.ffa.2014.10.001.

[9]

Y. Laigle-Chapuy, Permutation polynomial and applications to coding theory, Finite Fields Appl., 13 (2007), 58-70.  doi: 10.1016/j.ffa.2005.08.003.

[10]

K. LiL. Qu and X. Chen, New classes of permutation binomials and permutation trinomials over finite fields, Finite Fields Appl., 43 (2017), 69-85.  doi: 10.1016/j.ffa.2016.09.002.

[11]

K. LiL. QuC. Li and S. Fu, New permutation trinomials constructed from fractional polynomials, Acta Arith., 183 (2018), 101-116.  doi: 10.4064/aa8461-11-2017.

[12]

N. Li and T. Helleseth, Several classes of permutation trinomials from Niho exponents, Cryptogr. Commun., 9 (2017), 693-705.  doi: 10.1007/s12095-016-0210-9.

[13]

N. Li and T. Helleseth, New permutation trinomials from Niho exponents over finite fields with even characteristic, Cryptogr. Commun., 11 (2019), 129-136.  doi: 10.1007/s12095-018-0321-6.

[14]

N. Li and X. Zeng, A survey on the applications of Niho exponents, Cryptogr. Commun., 2018, 1–40. doi: 10.1007/s12095-018-0305-6.

[15]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. Cambridge University Press, 1997.

[16]

J. MaT. ZhangT. Feng and G. Ge, Some new results on permutation polynomials over finite fields, Des. Codes Cryptogr., 83 (2017), 425-443.  doi: 10.1007/s10623-016-0236-1.

[17]

Y. H. Park and J. B. Lee, Permutation polynomials and group permutation polynomials, Bull. Austral. Math. Soc., 63 (2001), 67-74.  doi: 10.1017/S0004972700019110.

[18]

R. L. RivestA. Shamir and L. M. Adelman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21 (1978), 120-126.  doi: 10.1145/359340.359342.

[19]

J. Schwenk and K. Huber, Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34 (1998), 759-760.  doi: 10.1049/el:19980569.

[20]

Z. Tu and X. Zeng, Two classes of permutation trinomials with Niho exponents, Finite Fields Appl., 53 (2018), 99-112.  doi: 10.1016/j.ffa.2018.05.007.

[21]

Z. TuX. Zeng and T. Helleseth, New permutation quadrinomials over $\mathbb{F}_{2^{2m}}$, Finite Fields Appl., 50 (2018), 304-318.  doi: 10.1016/j.ffa.2017.11.013.

[22]

Z. TuX. ZengC. Li and T. Helleseth, A class of new permutation trinomials, Finite Fields Appl., 50 (2018), 178-195.  doi: 10.1016/j.ffa.2017.11.009.

[23]

Z. TuX. Zeng and L. Hu, Several classes of complete permutation polynomials, Finite Fields Appl., 25 (2014), 182-193.  doi: 10.1016/j.ffa.2013.09.007.

[24]

Z. TuX. Zeng and Y. Jiang, Two classes of permutation polynomials having the form $(x^{2^m}+x+\delta)^s+x$, Finite Fields Appl., 31 (2015), 12-24.  doi: 10.1016/j.ffa.2014.09.005.

[25]

Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Lecture Notes in Comput. Sci., 4893 (2007), 119-128.  doi: 10.1007/978-3-540-77404-4_11.

[26]

D. WuP. YuanC. Ding and Y. Ma, Permutation trinomials over $\mathbb{F}_{2^m}$, Finite Fields Appl., 46 (2017), 38-56.  doi: 10.1016/j.ffa.2017.03.002.

[27]

Z. ZhaL. Hu and S. Fan, Further results on permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 45 (2017), 43-52.  doi: 10.1016/j.ffa.2016.11.011.

[28]

M. Zieve, On some permutation polynomials over $\mathbb{F}_q$ of the form $x^rh(x^{\frac{q-1}{d}})$, Proc. Amer. Math. Soc., 137 (2009), 2209-2216.  doi: 10.1090/S0002-9939-08-09767-0.

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