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Further results on permutation trinomials with Niho exponents

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Abstract

In this paper, we prove a conjecture proposed by Deng and Zheng about a class of permutation trinomials over finite fields \({\mathbb {F}}_{2^{2m}}\). In addition, we also construct four classes of permutation trinomials with Niho exponents over \({\mathbb {F}}_{3^{2m}}\).

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Acknowledgments

We would like to thank the editor and anonymous reviewers for their detailed and insightful comments, which have highly improved this paper. This work is partially supported by the Fundamental Research Funds for the Central Universities under Grant No. 21618331, the National Natural Science Foundation of China under Grant Nos. 61502482, 11871248, 11601462 and 61602125, the Natural Science Foundation of Guangxi under Grant No. 2016GXNSFBA380153, and the China Postdoctoral Science Foundation under Grant No. 2018M633041.

Author information

Authors and Affiliations

  1. College of Information Science and Technology, Jinan University, Guangzhou, 510632, China

    Libo Wang

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Baofeng Wu

  3. School of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Xiaoqiang Yue

  4. Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, 541004, China

    Yanbin Zheng

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  1. Libo Wang
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  2. Baofeng Wu
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  3. Xiaoqiang Yue
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  4. Yanbin Zheng
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Correspondence to Libo Wang.

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Appendix: MAGMA programs

Appendix: MAGMA programs

In this section we list some programs written in MAGMA which have been used in this paper.

1.1 A.1 Magma program for Theorem 1


F:=GF(32); K<x,y>:=PolynomialRing(F,2); Curve:=K!((x^11+x^7+x)*(y^10+y^4+1)+(y^11+y^7+y)*(x^10+x^4+1)); Factorization(Curve);

1.2 A.2 Magma program for Theorem 3


F:=GF(81); DefiningPolynomial(F); K<x,y>:=PolynomialRing(F,2); Curve:=K!((-x^6+x^4+1)*(y^7+y^3-y)-(-y^6+y^4+1)*(x^7+x^3-x)); Factorization(Curve); Hxy1:=Factorization(Curve)[6][1]; //Hxy2:=Factorization(Curve)[7][1]; //Hxy3:=Factorization(Curve)[8][1]; //Hxy4:=Factorization(Curve)[9][1]; Hxy1q:=K!(x*y*Evaluate(Hxy1,[1/x,1/y])); Resultant(Hxy1,Hxy1q,y); Factorization(Resultant(Hxy1,Hxy1q,y));

1.3 A.3 Magma program for Theorem 4


F:=GF(9); K<x,y>:=PolynomialRing(F,2); Curve:=K!((x^4+x^3-1)*(-y^5+y^2+y)-(y^4+y^3-1)*(-x^5+x^2+x)); Factorization(Curve); Hxy1:=Factorization(Curve)[2][1]; Hxy2:=Factorization(Curve)[3][1]; Resultant(Hxy1,Hxy2,y); Factorization(Resultant(Hxy1,Hxy2,y)); Resultant(Hxy1,Hxy2,x); Factorization(Resultant(Hxy1,Hxy2,x));

1.4 A.4 Magma program for Theorem 5


F:=GF(9); K<x,y>:=PolynomialRing(F,2); Curve:=K!((x^7+x^6-x)*(-y^6+y+1)-(y^7+y^6-y)*(-x^6+x+1)); Factorization(Curve); Hxy1:=Factorization(Curve)[2][1]; Hxy1q:=K!(x^3*y^3*Evaluate(Hxy1,[1/x,1/y])); Hxy2:=Factorization(Curve)[3][1]; Hxy2q:=K!(x^3*y^3*Evaluate(Hxy2,[1/x,1/y])); Resultant(Hxy1,Hxy1q,y); Factorization(Resultant(Hxy1,Hxy1q,y)); Resultant(Hxy2,Hxy2q,y); Factorization(Resultant(Hxy2,Hxy2q,y));

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Wang, L., Wu, B., Yue, X. et al. Further results on permutation trinomials with Niho exponents. Cryptogr. Commun. 11, 1057–1068 (2019). https://doi.org/10.1007/s12095-019-0349-2

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