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On some permutation trinomials in characteristic three

Year 2025, , 797 - 806, 24.06.2025

Burcu Gülmez Temür , Buket Özkaya

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

Abstract

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.

Keywords

permutation polynomials finite fields absolutely irreducible

References

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  • [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.
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  • [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.

Year 2025, , 797 - 806, 24.06.2025

Burcu Gülmez Temür , Buket Özkaya

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

Abstract

References

  • [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.
There are 35 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Mathematics
Authors

Burcu Gülmez Temür 0000-0002-0435-6894

Buket Özkaya 0000-0003-2658-5441

Early Pub Date August 27, 2024
Publication Date June 24, 2025
Submission Date February 27, 2024
Acceptance Date June 11, 2024
Published in Issue Year 2025

Cite

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. June 2025;54(3):797-806. doi:10.15672/hujms.1443686
Chicago Gülmez Temür, Burcu, and Buket Özkaya. “On Some Permutation Trinomials in Characteristic Three”. Hacettepe Journal of Mathematics and Statistics 54, no. 3 (June 2025): 797-806. https://doi.org/10.15672/hujms.1443686.
EndNote Gülmez Temür B, Özkaya B (June 1, 2025) On some permutation trinomials in characteristic three. Hacettepe Journal of Mathematics and Statistics 54 3 797–806.
IEEE B. Gülmez Temür and B. Özkaya, “On some permutation trinomials in characteristic three”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, pp. 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 (June 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 and Buket Özkaya. “On Some Permutation Trinomials in Characteristic Three”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 3, 2025, pp. 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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