doi: 10.3934/amc.2022045
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Compositional inverses of AGW-PPs –dedicated to professor cunsheng ding for his 60th birthday

School of Mathematics, South China Normal University, Guangzhou 510631, China

*Corresponding author: Pingzhi Yuan

Received  February 2022 Revised  April 2022 Early access June 2022

Fund Project: Supported by NSF of China grant 12171163

In this paper, we present a new method to obtain the compositional inverses of AGW-PPs. We improve some known results in this topic.

Citation: Pingzhi Yuan. Compositional inverses of AGW-PPs –dedicated to professor cunsheng ding for his 60th birthday. Advances in Mathematics of Communications, doi: 10.3934/amc.2022045
References:
[1]

A. AkbaryD. Ghioca and Q. Wang, On constructing permutations of finite fields, Finite Fields Their Appl., 17 (2011), 51-67.  doi: 10.1016/j.ffa.2010.10.002.

[2]

R. S. Coulter and M. Henderson, The compositional inverse of a class of permutation polynomials over a finite field, Bull. Austral. Math. Soc., 65 (2002), 521-526.  doi: 10.1017/S0004972700020578.

[3]

C. Ding, Cyclic codes from some monomials and trinomials, Siam J. Discrete Math., 27 (2013), 1977-1994.  doi: 10.1137/120882275.

[4]

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

[5]

C. DingQ. XiangJ. Yuan and P. Yuan, Explicit classes of permutation polynomials of $\mathbb{F}_{3^3m}$, Sci. China Ser. A, 52 (2009), 639-647.  doi: 10.1007/s11425-008-0142-8.

[6]

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.

[7]

C. Ding and Z. Zhou, Binary cyclic codes from explicit polynomials over $GF(2^m)$, Discrete Math., 321 (2014), 76-89.  doi: 10.1016/j.disc.2013.12.020.

[8]

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

[9]

K. LiL. Qu and Q. Wang, Compositional inverses of permutation polynomials of the form $x^rh(x^s)$ over finite fields, Cryptogr. Commun., 11 (2019), 279-298.  doi: 10.1007/s12095-018-0292-7.

[10] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. 
[11] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1986. 
[12]

G. L. Mullen, Permutation polynomials over finite fields, In: Proc. Conf. Finite Fields and Their Applications, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 141 (1993), 131–151.

[13]

T. NiuK. LiL. Qu and Q. Wang, Finding compositional inverses of permutations from the AGW creterion, IEEE Trans. Inf. Theory, 67 (2021), 4975-4985.  doi: 10.1109/TIT.2021.3089145.

[14]

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.

[15]

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

[16]

J. Schwenk and K. Huber, Public key encryption and digital signatures based on permutation polynomials, Electronic Letters, 34 (1998), 759-760. 

[17]

A. Tuxanidy and Q. Wang, On the inverses of some classes of permutations of finite fields, Finite Fields Their Appl., 28 (2014), 244-281.  doi: 10.1016/j.ffa.2014.02.006.

[18]

A. Tuxanidy and Q. Wang, Compositional inverses and complete mappings over finite fields, Discrete Appl. Math., 217 (2017), 318-329.  doi: 10.1016/j.dam.2016.09.009.

[19]

D. Wan and R. Lidl, Permutation polynomials of the form $x^rf(x^{(q-1)/d})$ and their group structure, Monatsh. Math., 112 (1991), 149-163.  doi: 10.1007/BF01525801.

[20]

Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Sequences, Subsequences, Consequences, 4893 (2007), 119-128.  doi: 10.1007/978-3-540-77404-4_11.

[21]

Q. Wang, A note on inverses of cyclotomic mapping permutation polynomials over finite fields, Finite Fields Their Appl., 45 (2017), 422-427.  doi: 10.1016/j.ffa.2017.01.006.

[22]

B. Wu, The compositional inverse of a class of linearized permutation polynomials over $ {{\mathbb F}}_{2^n}, n$ odd, Finite Fields Their Appl., 29 (2014), 34-48.  doi: 10.1016/j.ffa.2014.03.003.

[23]

B. Wu and Z. Liu, The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2, Finite Fields Their Appl., 24 (2013), 136-147.  doi: 10.1016/j.ffa.2013.05.003.

[24]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Their Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.

[25]

P. Yuan and C. Ding, Permutation polynomials over finite fields from a powerful lemma, Finite Fields Their Appl., 17 (2011), 560-574.  doi: 10.1016/j.ffa.2011.04.001.

[26]

P. Yuan and C. Ding, Further results on permutation polynomials over finite fields, Finite Fields Appl., 27 (2014), 88-103.  doi: 10.1016/j.ffa.2014.01.006.

[27]

P. Yuan and C. Ding, Permutation polynomials of the form $L(x)+S_2k^a+S_2k^b$ over $\mathbb{F}_{q^3k}$, Finite Fields Appl., 29 (2014), 106-117.  doi: 10.1016/j.ffa.2014.04.004.

[28]

D. ZhengM. YuanN. LiL. Hu and X. Zeng, Constructions of involutions over finite fields, IEEE Trans. Inf. Theory, 65 (2019), 7876-7883.  doi: 10.1109/TIT.2019.2919511.

[29]

Y. Zheng and Y. Yu, On inverse of permutation polynomials of small degree over finite fields II, preprint, 2018, arXiv: 1812.11812.

[30]

Y. ZhengQ. Wang and W. Wei, On inverses of permutation polynomials of small degree over finite fields, IEEE Trans. Inf. Theory, 66 (2020), 914-922.  doi: 10.1109/TIT.2019.2939113.

[31]

Y. ZhengY. YuY. Zhang and D. Pei, Piecewise constructions of inverses of cyclotomic mapping permutation polynomials, Finite Fields Appl., 40 (2016), 1-9.  doi: 10.1016/j.ffa.2016.02.005.

[32]

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

show all references

References:
[1]

A. AkbaryD. Ghioca and Q. Wang, On constructing permutations of finite fields, Finite Fields Their Appl., 17 (2011), 51-67.  doi: 10.1016/j.ffa.2010.10.002.

[2]

R. S. Coulter and M. Henderson, The compositional inverse of a class of permutation polynomials over a finite field, Bull. Austral. Math. Soc., 65 (2002), 521-526.  doi: 10.1017/S0004972700020578.

[3]

C. Ding, Cyclic codes from some monomials and trinomials, Siam J. Discrete Math., 27 (2013), 1977-1994.  doi: 10.1137/120882275.

[4]

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

[5]

C. DingQ. XiangJ. Yuan and P. Yuan, Explicit classes of permutation polynomials of $\mathbb{F}_{3^3m}$, Sci. China Ser. A, 52 (2009), 639-647.  doi: 10.1007/s11425-008-0142-8.

[6]

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.

[7]

C. Ding and Z. Zhou, Binary cyclic codes from explicit polynomials over $GF(2^m)$, Discrete Math., 321 (2014), 76-89.  doi: 10.1016/j.disc.2013.12.020.

[8]

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

[9]

K. LiL. Qu and Q. Wang, Compositional inverses of permutation polynomials of the form $x^rh(x^s)$ over finite fields, Cryptogr. Commun., 11 (2019), 279-298.  doi: 10.1007/s12095-018-0292-7.

[10] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. 
[11] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, Cambridge, 1986. 
[12]

G. L. Mullen, Permutation polynomials over finite fields, In: Proc. Conf. Finite Fields and Their Applications, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 141 (1993), 131–151.

[13]

T. NiuK. LiL. Qu and Q. Wang, Finding compositional inverses of permutations from the AGW creterion, IEEE Trans. Inf. Theory, 67 (2021), 4975-4985.  doi: 10.1109/TIT.2021.3089145.

[14]

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.

[15]

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

[16]

J. Schwenk and K. Huber, Public key encryption and digital signatures based on permutation polynomials, Electronic Letters, 34 (1998), 759-760. 

[17]

A. Tuxanidy and Q. Wang, On the inverses of some classes of permutations of finite fields, Finite Fields Their Appl., 28 (2014), 244-281.  doi: 10.1016/j.ffa.2014.02.006.

[18]

A. Tuxanidy and Q. Wang, Compositional inverses and complete mappings over finite fields, Discrete Appl. Math., 217 (2017), 318-329.  doi: 10.1016/j.dam.2016.09.009.

[19]

D. Wan and R. Lidl, Permutation polynomials of the form $x^rf(x^{(q-1)/d})$ and their group structure, Monatsh. Math., 112 (1991), 149-163.  doi: 10.1007/BF01525801.

[20]

Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Sequences, Subsequences, Consequences, 4893 (2007), 119-128.  doi: 10.1007/978-3-540-77404-4_11.

[21]

Q. Wang, A note on inverses of cyclotomic mapping permutation polynomials over finite fields, Finite Fields Their Appl., 45 (2017), 422-427.  doi: 10.1016/j.ffa.2017.01.006.

[22]

B. Wu, The compositional inverse of a class of linearized permutation polynomials over $ {{\mathbb F}}_{2^n}, n$ odd, Finite Fields Their Appl., 29 (2014), 34-48.  doi: 10.1016/j.ffa.2014.03.003.

[23]

B. Wu and Z. Liu, The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2, Finite Fields Their Appl., 24 (2013), 136-147.  doi: 10.1016/j.ffa.2013.05.003.

[24]

B. Wu and Z. Liu, Linearized polynomials over finite fields revisited, Finite Fields Their Appl., 22 (2013), 79-100.  doi: 10.1016/j.ffa.2013.03.003.

[25]

P. Yuan and C. Ding, Permutation polynomials over finite fields from a powerful lemma, Finite Fields Their Appl., 17 (2011), 560-574.  doi: 10.1016/j.ffa.2011.04.001.

[26]

P. Yuan and C. Ding, Further results on permutation polynomials over finite fields, Finite Fields Appl., 27 (2014), 88-103.  doi: 10.1016/j.ffa.2014.01.006.

[27]

P. Yuan and C. Ding, Permutation polynomials of the form $L(x)+S_2k^a+S_2k^b$ over $\mathbb{F}_{q^3k}$, Finite Fields Appl., 29 (2014), 106-117.  doi: 10.1016/j.ffa.2014.04.004.

[28]

D. ZhengM. YuanN. LiL. Hu and X. Zeng, Constructions of involutions over finite fields, IEEE Trans. Inf. Theory, 65 (2019), 7876-7883.  doi: 10.1109/TIT.2019.2919511.

[29]

Y. Zheng and Y. Yu, On inverse of permutation polynomials of small degree over finite fields II, preprint, 2018, arXiv: 1812.11812.

[30]

Y. ZhengQ. Wang and W. Wei, On inverses of permutation polynomials of small degree over finite fields, IEEE Trans. Inf. Theory, 66 (2020), 914-922.  doi: 10.1109/TIT.2019.2939113.

[31]

Y. ZhengY. YuY. Zhang and D. Pei, Piecewise constructions of inverses of cyclotomic mapping permutation polynomials, Finite Fields Appl., 40 (2016), 1-9.  doi: 10.1016/j.ffa.2016.02.005.

[32]

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

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