Abstract
Let \({\mathbb {F}}_q\) denote the finite field of order q. In this paper, we determine certain permutation binomials and permutation trinomials of the form \(x^{r}h(x^{q+1})\) over \(\mathbb {F}_{q^2}\). Some of them are generalizations of known ones.
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References
Bassalygo, L.A., Zinoviev, V.A.: Permutation and complete permutation polynomials. Finite Fields Appl. 33, 198–211 (2015)
Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, part I. Ann. Math. 11, 65–120 (1896–1897)
Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59, 5898–5904 (2013)
Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006)
Fernando, N., Hou, X., Lappano, S.D.: Permutation polynomials over finite fields involving \(x+x^q+\dots +x^{q^{a-1}}\). Discrete Math. 315–316, 173–184 (2014)
Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016)
Gupta, R., Sharma, R.K.: Further results on permutation polynomials of the form \((x^{p^m}-x+\delta )^s+x\) over \(\mathbb{F}_{p^{2m}}\). Finite Fields Appl. 50, 196–208 (2018)
Hou, X.: A survey of permutation binomials and trinomials over finite fields. In: Proceedings of the 11th International Conference on Finite Fields and Their Applications, Contemp. Math., Magdeburg, Germany, July 2013, vol. 632, pp. 177–191. AMS (2015)
Hou, X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015)
Hou, X.: Determination of a type of permutation trinomials over finite fields. Acta Arith. 166, 253–278 (2014)
Hou, X.: Determination of a type of permutation trinomials over finite fields II. Finite Fields Appl. 35, 16–35 (2015)
Hou, X., Lappano, S.D.: Determination of a type of permutation binomials over finite fields. J. Number Theory 147, 14–23 (2015)
Laigle-Chapuy, Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007)
Li, J., Chandler, D.B., Xiang, Q.: Permutation polynomials of degree \(6\) or \(7\) over finite fields of characterstic \(2\). Finite Fields Appl. 16, 406–419 (2010)
Li, N., Helleseth, T., Tang, X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013)
Lidl, R., Mullen, W.B.: Permutation Polynomials in RSA-Cryptosystems. Advances in Cryptology, pp. 293–301. Plenum, New York (1984)
Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge Univ. Press, Cambridge (1997)
Marcos, J.E.: Some permutation polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 26, 465–474 (2015)
Schwenk, J., Huber, K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34, 759–760 (1998)
Shallue, C.J., Wanless, I.M.: Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields Appl. 20, 84–92 (2013)
Wang, Q.: Cyclotomy and permutation polynomials of large indices. Finite Fields Appl. 22, 57–69 (2013)
Yuan, P., Ding, C.: Further results on permutation polynomials over finite fields. Finite Fields Appl. 27, 88–103 (2014)
Yuan, P., Ding, C.: Permutation polynomials of the form \(L(x)+S_{2k}^a+S_{2k}^b\) over \(\mathbb{F}_{q^{3k}}\). Finite Fields Appl. 29, 106–117 (2014)
Zieve, M.E.: Some families of permutation polynomials over finite fields. Int. J. Number Theory 4, 851–857 (2008)
Zieve, M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal Latin squares. arXiv:1312.1325
Acknowledgements
The authors gratefully acknowledge the valuable suggestions from the anonymous referees which improved the quality and presentation of the paper. The first author is ConsenSys Block Chain Chair Professor. He wants to thank the CosenSys AG for the same. The second author wants to thank for its support, CSIR, New Delhi, Govt. of India, Grant No. F.No. 09/086(1135)/2012-EMR-I.
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Appendix
Appendix
We append here the effectively computable sets mentioned in Sect. 4. The computations to obtain these sets are carried out with MAGMA online.
1.1 Computed set for Theorem 4.7
For \(q=5\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if either \(a=\pm 1\) and \(b^2=\pm 2\), or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{5^2}\) given by \(x^2+x+2\) and \((k,l) \in S_1\) are pairs of positive integers modulo \(24\}\) where \(S_1\) is given by
1.2 Computed set for Theorem 4.8
For \(q=7\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if the conditions of Theorem 4.8(i) or Theorem 4.8(iv) holds, or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{7^2}\) given by \(x^2+x+3\) and \((k,l) \in S_2 \cup S_3\) are pairs of positive integers modulo \(48\}\) where \(S_2\) and \(S_3\) are given by
For \(q=13\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if the conditions of Theorem 4.8(i) or Theorem 4.8(iv) holds, or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{13^2}\) given by \(x^2+x+2\) and \((k,l) \in S_4\) are pairs of positive integers modulo \(168\}\) where \(S_4\) is given by
1.3 Computed set for Theorem 4.11
For \(q=8\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if the conditions of Theorem 4.11(iii) holds or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{2^6}\) given by \(x^6+x+1\) and \((k,l) \in S_5\) are pairs of positive integers modulo \(63\}\) where \(S_5=T_1 \cup T_2 \cup T_3 \cup T_4\), and \(T_i\) for \(1\le i \le 4\) are given by
and
For \(q=16\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if the conditions of Theorem 4.11(iii) holds or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{2^8}\) given by \(x^8+x^4+x^3+x^2+1\) and \((k,l) \in S_6\) are pairs of positive integers modulo \(255\}\) where \(S_6\) is given by
For \(q=32\), the polynomial \(f(x) = ax + bx^{q+2} + x^{2q+3}\) permutes \(\mathbb {F}_{q^2}\) if and only if the conditions of Theorem 4.11(iii) holds or \((a,b)\in \{(\alpha ^k,\alpha ^l) ~| ~\alpha \) is a primitive element of \(\mathbb {F}_{2^{10}}\) given by \(x^{10}+x^4+x^3+x^2+1\) and \((k,l) \in S_7\) are pairs of positive integers modulo \(1023\}\) where \(S_7\) is given by
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Sharma, R.K., Gupta, R. Determination of a type of permutation binomials and trinomials. AAECC 31, 65–86 (2020). https://doi.org/10.1007/s00200-019-00394-y
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DOI: https://doi.org/10.1007/s00200-019-00394-y