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On some permutation trinomials in characteristic three

Yıl 2025, Cilt: 54 Sayı: 3, 797 - 806, 24.06.2025

Burcu Gülmez Temür , Buket Özkaya

https://doi.org/10.15672/hujms.1443686

Öz

In this paper, we determine the permutation properties of the polynomial $x^3+x^{q+2}-x^{4q-1}$ over the finite field $\mathbb{F}_{q^2}$ in characteristic three. Moreover, we consider the trinomials of the form $x^{4q-1}+x^{2q+1} \pm x^{3}$. In particular, we first show that $x^3+x^{q+2}-x^{4q-1}$ permutes $\mathbb{F}_{q^2}$ with $q=3^m$ if and only if $m$ is odd. This enables us to show that the sufficient condition in [34, Theorem 4] is also necessary. Next, we prove that $x^{4q-1}+x^{2q+1} - x^{3}$ permutes $\mathbb{F}_{q^2}$ with $q=3^m$ if and only if $m\not\equiv 0 \pmod 4$. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial $x^{4q-1}+x^{2q+1} + x^{3}$ and show that it is never a permutation polynomial of $\mathbb{F}_{q^2}$ in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.

Anahtar Kelimeler

permutation polynomials finite fields absolutely irreducible

Kaynakça

  • [1] A. Akbary and Q. Wang, On polynomials of the form $x^rf(x^{(q-1)/l})$, Int. J. Math. Math. Sci., Art. ID 23408, 2007.
  • [2] T. Bai and Y. Xia, A new class of permutation trinomials constructed from Niho exponents, Cryptogr. Commun. 10, 1023-1036, 2018.
  • [3] D. Bartoli and M. Giulietti, Permutation polynomials, fractional polynomials, and algebraic curves, Finite Fields Appl. 51, 1-16, 2018.
  • [4] D. Bartoli and M. Timpanella, A family of permutation trinomials over $\mathbb{F}_{q^2}$, Finite Fields Appl. 70, 101781, 2021.
  • [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24, 1179-1260, 1997.
  • [6] X. Cao, X. Hou, J. Mi and S. Xu, More permutation polynomials with Niho exponents which permute $\mathbb{F}_{q^2}$ , Finite Fields Appl. 62, 101626, 2020.
  • [7] D. Cox, D. Little and D. O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer, Cham, 2015.
  • [8] H. Deng and D. Zheng, More classes of permutation trinomials with Niho exponents, Cryptogr. Commun. 11, 227-236, 2019.
  • [9] L. E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math. 11, 65-120, 1896.
  • [10] M. Grassl, F. Özbudak, B. Özkaya and B. Gülmez Temür, Complete Characterization of a Class of Permutation Trinomial in Characteristic Five, to appear in Cryptogr. Commun., DOI: |https://doi.org/10.1007/s12095-024-00705-2.
  • [11] R. Gupta and R. K. Sharma, Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields Appl. 41, 89-96, 2016.
  • [12] C. Hermite, Sur les fonctions de sept lettres, C.R. Acad. Sci. Paris 57, 750-757, 1863.
  • [13] X. Hou, Permutation polynomials over finite fields - a survey of recent advances, Finite Fields Appl. 32, 82-119, 2015.
  • [14] X. Hou, Determination of a type of permutation trinomials over finite fields, Acta Arith. 166 (3), 253-278, 2014.
  • [15] X. Hou, Determination of a type of permutation trinomials over finite fields, II, Finite Fields Appl. 35, 16-35, 2015.
  • [16] X. Hou, A survey of permutation binomials and trinomials over finite fields (English summary), Topics in finite fields, 177-191, Contemp. Math. 632, Amer. Math. Soc., Providence, RI, 2015.
  • [17] X. Hou, Lectures on finite fields, Graduate Studies in Mathematics, 190, American Mathematical Society, Providence, RI, 2018.
  • [18] L. Li, C. Li, C. Li and X. Zeng, New classes of complete permutation polynomials, Finite Fields Appl. 55, 177-201, 2019.
  • [19] K. Li, L. Qu and X. Chen, New classes of permutation binomials and permutation trinomials over finite fields, Finite Fields Appl. 43, 69-85, 2017.
  • [20] K. Li, L. Qu, C. Li and S. Fu, New Permutation Trinomials Constructed from Fractional Polynomials, Acta Arith. 183, 101-116, 2018.
  • [21] K. Li, L. Qu and Q. Wang, New constructions of permutation polynomials of the form $x^rh(x^{q-1})$ over $\mathbb{F}_{q^2}$, Des. Codes Cryptogr. 86, 2379-2405, 2018.
  • [22] L. Li, Q. Wang, Y. Xu and X. Zeng, Several classes of complete permutation polynomials with Niho exponents, Finite Fields Appl. 72, 101831, 2021.
  • [23] R. Lidl and H. Niederreiter, Finite Fields, (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Cambridge, 1997.
  • [24] G. L. Mullen and D. Panario, Handbook of Finite Fields, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2013.
  • [25] F. Özbudak and B. Gülmez Temür, Classification of permutation polynomials of the form $x^3g(x^{q-1})$ of $\mathbb{F}_{q^2}$ where $g(x)=x^3+bx+c$ and $b,c \in \mathbb{F}_q^*$ , Des. Codes Cryptogr. 90, 1537-1556, 2022.
  • [26] F. Özbudak and B. Gülmez Temür, Complete characterization of some permutation polynomials of the form $x^r(1+ax^{s_1(q-1)}+bx^{s_2(q-1)})$ over $\mathbb{F}_{q^2}$ , Cryptogr. Commun. 15, 775-793, 2023.
  • [27] F. Özbudak and B. Gülmez Temür, Classification of some quadrinomials over finite fields of odd characteristic, Finite Fields Appl. 87, 102158, 2023.
  • [28] Y. H. Park and J. B. Lee, Permutation polynomials and group permutation polynomials, Bull. Austral. Math. Soc. 63, 67-74, 2001.
  • [29] Z. Tu and X. Zeng, A class of permutation trinomials over finite fields of odd characteristic, Cryptogr. Commun. 11, 563-583, 2019.
  • [30] Z. Tu, X. Zeng, C. Li and T. Helleseth, A class of new permutation trinomials, Finite Fields Appl. 50, 178-195, 2018.
  • [31] D. Wan and R. Lidl, Permutation polynomials of the form $x^rf(x^{(q-1)/d})$ and their group structure, Monatshefte Math. 112, 149-163, 1991.
  • [32] Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Sequences, subsequences, and consequences, Lecture Notes in Comput. Sci. 4893, Springer, Berlin, 119-128, 2007.
  • [33] Q. Wang, Polynomials over finite fields: an index approach, Combinatorics and Finite Fields, Difference Sets, Polynomials, Pseudorandomness and Applications, De Gruyter, 319-348, 2019.
  • [34] L. Wang, B. Wu, X. Yue and Y. Zheng, Further results on permutation trinomials with Niho exponents, Cryptogr. Commun. 11, 1057-1068, 2019.
  • [35] M. E. Zieve, On some permutation polynomials over $\mathbb{F}_q$ of the form $x^rh(x^{(q-1)/d})$, Proc. Amer. Math. Soc. 137, 2209-2216, 2009.

Yıl 2025, Cilt: 54 Sayı: 3, 797 - 806, 24.06.2025

Burcu Gülmez Temür , Buket Özkaya

https://doi.org/10.15672/hujms.1443686

Öz

Kaynakça

  • [1] A. Akbary and Q. Wang, On polynomials of the form $x^rf(x^{(q-1)/l})$, Int. J. Math. Math. Sci., Art. ID 23408, 2007.
  • [2] T. Bai and Y. Xia, A new class of permutation trinomials constructed from Niho exponents, Cryptogr. Commun. 10, 1023-1036, 2018.
  • [3] D. Bartoli and M. Giulietti, Permutation polynomials, fractional polynomials, and algebraic curves, Finite Fields Appl. 51, 1-16, 2018.
  • [4] D. Bartoli and M. Timpanella, A family of permutation trinomials over $\mathbb{F}_{q^2}$, Finite Fields Appl. 70, 101781, 2021.
  • [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24, 1179-1260, 1997.
  • [6] X. Cao, X. Hou, J. Mi and S. Xu, More permutation polynomials with Niho exponents which permute $\mathbb{F}_{q^2}$ , Finite Fields Appl. 62, 101626, 2020.
  • [7] D. Cox, D. Little and D. O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer, Cham, 2015.
  • [8] H. Deng and D. Zheng, More classes of permutation trinomials with Niho exponents, Cryptogr. Commun. 11, 227-236, 2019.
  • [9] L. E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math. 11, 65-120, 1896.
  • [10] M. Grassl, F. Özbudak, B. Özkaya and B. Gülmez Temür, Complete Characterization of a Class of Permutation Trinomial in Characteristic Five, to appear in Cryptogr. Commun., DOI: |https://doi.org/10.1007/s12095-024-00705-2.
  • [11] R. Gupta and R. K. Sharma, Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields Appl. 41, 89-96, 2016.
  • [12] C. Hermite, Sur les fonctions de sept lettres, C.R. Acad. Sci. Paris 57, 750-757, 1863.
  • [13] X. Hou, Permutation polynomials over finite fields - a survey of recent advances, Finite Fields Appl. 32, 82-119, 2015.
  • [14] X. Hou, Determination of a type of permutation trinomials over finite fields, Acta Arith. 166 (3), 253-278, 2014.
  • [15] X. Hou, Determination of a type of permutation trinomials over finite fields, II, Finite Fields Appl. 35, 16-35, 2015.
  • [16] X. Hou, A survey of permutation binomials and trinomials over finite fields (English summary), Topics in finite fields, 177-191, Contemp. Math. 632, Amer. Math. Soc., Providence, RI, 2015.
  • [17] X. Hou, Lectures on finite fields, Graduate Studies in Mathematics, 190, American Mathematical Society, Providence, RI, 2018.
  • [18] L. Li, C. Li, C. Li and X. Zeng, New classes of complete permutation polynomials, Finite Fields Appl. 55, 177-201, 2019.
  • [19] K. Li, L. Qu and X. Chen, New classes of permutation binomials and permutation trinomials over finite fields, Finite Fields Appl. 43, 69-85, 2017.
  • [20] K. Li, L. Qu, C. Li and S. Fu, New Permutation Trinomials Constructed from Fractional Polynomials, Acta Arith. 183, 101-116, 2018.
  • [21] K. Li, L. Qu and Q. Wang, New constructions of permutation polynomials of the form $x^rh(x^{q-1})$ over $\mathbb{F}_{q^2}$, Des. Codes Cryptogr. 86, 2379-2405, 2018.
  • [22] L. Li, Q. Wang, Y. Xu and X. Zeng, Several classes of complete permutation polynomials with Niho exponents, Finite Fields Appl. 72, 101831, 2021.
  • [23] R. Lidl and H. Niederreiter, Finite Fields, (Encyclopedia of Mathematics and its Applications), Cambridge University Press, Cambridge, 1997.
  • [24] G. L. Mullen and D. Panario, Handbook of Finite Fields, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2013.
  • [25] F. Özbudak and B. Gülmez Temür, Classification of permutation polynomials of the form $x^3g(x^{q-1})$ of $\mathbb{F}_{q^2}$ where $g(x)=x^3+bx+c$ and $b,c \in \mathbb{F}_q^*$ , Des. Codes Cryptogr. 90, 1537-1556, 2022.
  • [26] F. Özbudak and B. Gülmez Temür, Complete characterization of some permutation polynomials of the form $x^r(1+ax^{s_1(q-1)}+bx^{s_2(q-1)})$ over $\mathbb{F}_{q^2}$ , Cryptogr. Commun. 15, 775-793, 2023.
  • [27] F. Özbudak and B. Gülmez Temür, Classification of some quadrinomials over finite fields of odd characteristic, Finite Fields Appl. 87, 102158, 2023.
  • [28] Y. H. Park and J. B. Lee, Permutation polynomials and group permutation polynomials, Bull. Austral. Math. Soc. 63, 67-74, 2001.
  • [29] Z. Tu and X. Zeng, A class of permutation trinomials over finite fields of odd characteristic, Cryptogr. Commun. 11, 563-583, 2019.
  • [30] Z. Tu, X. Zeng, C. Li and T. Helleseth, A class of new permutation trinomials, Finite Fields Appl. 50, 178-195, 2018.
  • [31] D. Wan and R. Lidl, Permutation polynomials of the form $x^rf(x^{(q-1)/d})$ and their group structure, Monatshefte Math. 112, 149-163, 1991.
  • [32] Q. Wang, Cyclotomic mapping permutation polynomials over finite fields, Sequences, subsequences, and consequences, Lecture Notes in Comput. Sci. 4893, Springer, Berlin, 119-128, 2007.
  • [33] Q. Wang, Polynomials over finite fields: an index approach, Combinatorics and Finite Fields, Difference Sets, Polynomials, Pseudorandomness and Applications, De Gruyter, 319-348, 2019.
  • [34] L. Wang, B. Wu, X. Yue and Y. Zheng, Further results on permutation trinomials with Niho exponents, Cryptogr. Commun. 11, 1057-1068, 2019.
  • [35] M. E. Zieve, On some permutation polynomials over $\mathbb{F}_q$ of the form $x^rh(x^{(q-1)/d})$, Proc. Amer. Math. Soc. 137, 2209-2216, 2009.
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Matematik
Yazarlar

Burcu Gülmez Temür 0000-0002-0435-6894

Buket Özkaya 0000-0003-2658-5441

Erken Görünüm Tarihi 27 Ağustos 2024
Yayımlanma Tarihi 24 Haziran 2025
Gönderilme Tarihi 27 Şubat 2024
Kabul Tarihi 11 Haziran 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 54 Sayı: 3

Kaynak Göster

APA Gülmez Temür, B., & Özkaya, B. (2025). On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics, 54(3), 797-806. https://doi.org/10.15672/hujms.1443686
AMA Gülmez Temür B, Özkaya B. On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics. Haziran 2025;54(3):797-806. doi:10.15672/hujms.1443686
Chicago Gülmez Temür, Burcu, ve Buket Özkaya. “On Some Permutation Trinomials in Characteristic Three”. Hacettepe Journal of Mathematics and Statistics 54, sy. 3 (Haziran 2025): 797-806. https://doi.org/10.15672/hujms.1443686.
EndNote Gülmez Temür B, Özkaya B (01 Haziran 2025) On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics 54 3 797–806.
IEEE B. Gülmez Temür ve B. Özkaya, “On some permutation trinomials in characteristic three”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, ss. 797–806, 2025, doi: 10.15672/hujms.1443686.
ISNAD Gülmez Temür, Burcu - Özkaya, Buket. “On Some Permutation Trinomials in Characteristic Three”. Hacettepe Journal of Mathematics and Statistics 54/3 (Haziran 2025), 797-806. https://doi.org/10.15672/hujms.1443686.
JAMA Gülmez Temür B, Özkaya B. On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics. 2025;54:797–806.
MLA Gülmez Temür, Burcu ve Buket Özkaya. “On Some Permutation Trinomials in Characteristic Three”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, 2025, ss. 797-06, doi:10.15672/hujms.1443686.
Vancouver Gülmez Temür B, Özkaya B. On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics. 2025;54(3):797-806.

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