# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022046
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## Two classes of power mappings with boomerang uniformity 2

 1 School of Mathematics, Southwest Jiaotong University, China 2 The Experimental High School Attached to Beijing Normal University, China 3 School of Mathematical Sciences, Peking University, China

*Corresponding author: Haode Yan (hdyan@swjtu.edu.cn)

Received  March 2022 Revised  May 2022 Early access June 2022

Fund Project: H. Yan's research was supported by the National Natural Science Foundation of China (Grant No.11801468) and the Fundamental Research Funds for the Central Universities of China (Grant No.2682021ZTPY076)

Let $q$ be an odd prime power. Let $F_1(x) = x^{d_1}$ and $F_2(x) = x^{d_2}$ be power mappings over $\mathrm{GF}(q^2)$, where $d_1 = q-1$ and $d_2 = d_1+\frac{q^2-1}{2} = \frac{(q-1)(q+3)}{2}$. In this paper, we study the boomerang uniformity of $F_1$ and $F_2$ via their differential properties. It is shown that the boomerang uniformity of $F_i$ ($i = 1,2$) is 2 with some conditions on $q$.

Citation: Haode Yan, Zhen Li, Zhitian Song, Rongquan Feng. Two classes of power mappings with boomerang uniformity 2. Advances in Mathematics of Communications, doi: 10.3934/amc.2022046
##### References:
 [1] E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563. [2] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of power functions, Int. J. Inf. Coding Theory, 1 (2010), 149-170.  doi: 10.1504/IJICOT.2010.032132. [3] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of ${x\mapsto x^{2^{t}-1}}$, IEEE Trans. Inform. Theory, 57 (2011), 8127-8137.  doi: 10.1109/TIT.2011.2169129. [4] C. Boura and A. Canteaut, On the boomerang uniformity of cryptographic sboxes, IACR Trans. Symmetric Cryptology, 2018 (2018), 290–310. doi: 10.13154/tosc.v2018.i3.290-310. [5] K. Browning, J. Dillon, M. McQuistan and J. Wolfe, An APN permutation in dimension six, Finite Fields: Theory and Applications, Contemp. Math. Amer. Math. Soc., 518 (2010), 33–42. doi: 10.1090/conm/518/10194. [6] L. Budaghyan and T. Helleseth, New perfect nonlinear multinomials over $F_{p^2k}$ for any odd prime $p$, Sequences and Their Applications—SETA 2008, Lecture Notes in Comput. Sci., 5203 (2008), 403–414. doi: 10.1007/978-3-540-85912-3_35. [7] M. Calderini and I. Villa, On the boomerang uniformity of some permutation polynomials, Cryptogr. Commun., 12 (2020), 1161-1178.  doi: 10.1007/s12095-020-00439-x. [8] S. Choi, S. Hong, J. No and H. Chung, Differential spectrum of some power functions in odd prime characteristic, Finite Fields Appl., 21 (2013), 11-29.  doi: 10.1016/j.ffa.2013.01.002. [9] C. Cid, T. Huang, T. Peyrin, Y. Sasaki and L. Song, Boomerang connectivity table: A new cryptanalysis tool, Advances in Cryptology—EUROCRYPT 2018, 10821 (2018), 683–714. doi: 10.1007/978-3-319-78375-8_22. [10] R. Coulter and R. Matthews, Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10 (1997), 167-184.  doi: 10.1023/A:1008292303803. [11] C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combin. Theory Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006. [12] H. Dobbertin, Almost perfect nonlinear power functions on GF($2^n$): The Welch case, IEEE Trans. Inform. Theory, 45 (1999), 1271-1275.  doi: 10.1109/18.761283. [13] H. Dobbertin, Almost perfect nonlinear power functions on $\rm GF(2^n)$: The Niho case, Inform. and Comput., 151 (1999), 57-72.  doi: 10.1006/inco.1998.2764. [14] H. Dobbertin, T. Helleseth, P. Kumar and H. Martinsen, Ternary m-sequences with three-valued cross-correlation function :New decimations of Welch and Niho type, IEEE Trans. Inform. Theory, 47 (2001), 1473-1481.  doi: 10.1109/18.923728. [15] H. Dobbertin, D. Mills, E. Müller, A. Pott and W. Willems, APN functions in odd characteristic, Discrete Math., 267 (2003), 95-112.  doi: 10.1016/S0012-365X(02)00606-4. [16] S. Hasan, M. Pal and P. Stǎnicǎ, Boomerang uniformity of a class of power maps, Des. Codes Cryptogr., 89 (2021), 2627-2636.  doi: 10.1007/s10623-021-00944-x. [17] T. Helleseth, C. Rong and D. Sandberg, New families of almost perfect nonlinear power mappings, IEEE Trans. Inform. Theory, 45 (1999), 474-485.  doi: 10.1109/18.748997. [18] T. Helleseth and D. Sandberg, Some power mappings with low differential uniformity, Appl. Algebra Engrg. Comm. Comput., 8 (1997), 363-370.  doi: 10.1007/s002000050073. [19] S. Jiang, K. Li, Y. Li and L. Qu, Differential and boomerang spectrums of some power permutations, Cryptogr. Commun., 14 (2022), 371-393.  doi: 10.1007/s12095-021-00530-x. [20] K. Li, L. Qu, B. Sun and C. Li, New results about the boomerang uniformity of permutation polynomials, IEEE Trans. Inform. Theory, 65 (2019), 7542-7553.  doi: 10.1109/TIT.2019.2918531. [21] N. Li, Y. Wu, X. Zeng and X. Tang, On the differential spectrum of a class of power functions over finite fields, arXiv: 2012.04316. [22] Z. Li and H. Yan, Differential spectra of a class of power permutations with niho exponents, Adv. Math. Commun., 2021. doi: 10.3934/amc.2021060. [23] S. Mesnager, B. Mandal and M. Msahli, Survey on recent trends towards generalized differential and boomerang uniformities, Cryptogr. Commun., 14 (2022), 691-735.  doi: 10.1007/s12095-021-00551-6. [24] G. Ness and T. Helleseth, A new family of ternary almost perfect nonlinear mappings, IEEE Trans. Inform. Theory, 53 (2007), 2581-2586.  doi: 10.1109/TIT.2007.899508. [25] K. Nyberg, Differentially uniform mappings for cryptography, Advances in Cryptology—EUROCRYPT '93 (Lofthus, 1993), Lecture Notes in Comput. Sci., 765 (1994), 55–64. doi: 10.1007/3-540-48285-7_6. [26] D. Wagner, The boomerang attack, Fast Software Encryption, Lecture Notes in Comput. Sci., 1636 (1999), 156-170.  doi: 10.1007/3-540-48519-8_12. [27] Y. Xia, X. Zhang, C. Li and T. Helleseth, The differential spectrum of a ternary power mapping, Finite Fields Appl., 64 (2020), 101660, 16 pp. doi: 10.1016/j.ffa.2020.101660. [28] M. Xiong and H. Yan, A note on the differential spectrum of a 4-uniform power function, Finite Fields Appl., 48 (2017), 117-125.  doi: 10.1016/j.ffa.2017.07.008. [29] M. Xiong, H. Yan and P. Yuan, On a conjecture of differentially 8-uniform power function, Des. Codes Cryptogr., 86 (2018), 1601-1621.  doi: 10.1007/s10623-017-0416-7. [30] H. Yan and C. Li, Differential spectra of a class of power permutations with characteristic 5, Des. Codes Cryptogr., 89 (2021), 1181-1191.  doi: 10.1007/s10623-021-00865-9. [31] H. Yan, Y. Xia, C. Li, T. Helleseth, M. Xiong and J. Luo, The differential spectrum of the power mapping $x^{p^n-3}$, IEEE Trans. Inform. Theory. [32] H. Yan, Z. Zhou, J. Weng, J. Wen, T. Helleseth and Q. Wang, Differential spectrum of Kasami power permutations over odd characteristic finite fields, IEEE Trans. Inform. Theory, 65 (2019), 6819-6826.  doi: 10.1109/TIT.2019.2910070. [33] Z. Zha and L. Hu, The boomerang uniformity of power permutations $x^{2^k-1}$ over $F_{2^n}$, Ninth International Workshop on Signal Design and its Applications in Communications (IWSDA), (2019). doi: 10.1109/IWSDA46143.2019.8966114. [34] Z. Zha, G. Kyureghyan and X. Wang, Perfect nonlinear binomials and their semifields, Finite Fields Appl., 15 (2009), 125-133.  doi: 10.1016/j.ffa.2008.09.002. [35] Z. Zha and X. Wang, Almost perfect nonlinear power functions in odd characteristic, IEEE Trans. Inform. Theory, 57 (2011), 4826-4832.  doi: 10.1109/TIT.2011.2145130.

show all references

##### References:
 [1] E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563. [2] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of power functions, Int. J. Inf. Coding Theory, 1 (2010), 149-170.  doi: 10.1504/IJICOT.2010.032132. [3] C. Blondeau, A. Canteaut and P. Charpin, Differential properties of ${x\mapsto x^{2^{t}-1}}$, IEEE Trans. Inform. Theory, 57 (2011), 8127-8137.  doi: 10.1109/TIT.2011.2169129. [4] C. Boura and A. Canteaut, On the boomerang uniformity of cryptographic sboxes, IACR Trans. Symmetric Cryptology, 2018 (2018), 290–310. doi: 10.13154/tosc.v2018.i3.290-310. [5] K. Browning, J. Dillon, M. McQuistan and J. Wolfe, An APN permutation in dimension six, Finite Fields: Theory and Applications, Contemp. Math. Amer. Math. Soc., 518 (2010), 33–42. doi: 10.1090/conm/518/10194. [6] L. Budaghyan and T. Helleseth, New perfect nonlinear multinomials over $F_{p^2k}$ for any odd prime $p$, Sequences and Their Applications—SETA 2008, Lecture Notes in Comput. Sci., 5203 (2008), 403–414. doi: 10.1007/978-3-540-85912-3_35. [7] M. Calderini and I. Villa, On the boomerang uniformity of some permutation polynomials, Cryptogr. Commun., 12 (2020), 1161-1178.  doi: 10.1007/s12095-020-00439-x. [8] S. Choi, S. Hong, J. No and H. Chung, Differential spectrum of some power functions in odd prime characteristic, Finite Fields Appl., 21 (2013), 11-29.  doi: 10.1016/j.ffa.2013.01.002. [9] C. Cid, T. Huang, T. Peyrin, Y. Sasaki and L. Song, Boomerang connectivity table: A new cryptanalysis tool, Advances in Cryptology—EUROCRYPT 2018, 10821 (2018), 683–714. doi: 10.1007/978-3-319-78375-8_22. [10] R. Coulter and R. Matthews, Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10 (1997), 167-184.  doi: 10.1023/A:1008292303803. [11] C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combin. Theory Ser. A, 113 (2006), 1526-1535.  doi: 10.1016/j.jcta.2005.10.006. [12] H. Dobbertin, Almost perfect nonlinear power functions on GF($2^n$): The Welch case, IEEE Trans. Inform. Theory, 45 (1999), 1271-1275.  doi: 10.1109/18.761283. [13] H. Dobbertin, Almost perfect nonlinear power functions on $\rm GF(2^n)$: The Niho case, Inform. and Comput., 151 (1999), 57-72.  doi: 10.1006/inco.1998.2764. [14] H. Dobbertin, T. Helleseth, P. Kumar and H. Martinsen, Ternary m-sequences with three-valued cross-correlation function :New decimations of Welch and Niho type, IEEE Trans. Inform. Theory, 47 (2001), 1473-1481.  doi: 10.1109/18.923728. [15] H. Dobbertin, D. Mills, E. Müller, A. Pott and W. Willems, APN functions in odd characteristic, Discrete Math., 267 (2003), 95-112.  doi: 10.1016/S0012-365X(02)00606-4. [16] S. Hasan, M. Pal and P. Stǎnicǎ, Boomerang uniformity of a class of power maps, Des. Codes Cryptogr., 89 (2021), 2627-2636.  doi: 10.1007/s10623-021-00944-x. [17] T. Helleseth, C. Rong and D. Sandberg, New families of almost perfect nonlinear power mappings, IEEE Trans. Inform. Theory, 45 (1999), 474-485.  doi: 10.1109/18.748997. [18] T. Helleseth and D. Sandberg, Some power mappings with low differential uniformity, Appl. Algebra Engrg. Comm. Comput., 8 (1997), 363-370.  doi: 10.1007/s002000050073. [19] S. Jiang, K. Li, Y. Li and L. Qu, Differential and boomerang spectrums of some power permutations, Cryptogr. Commun., 14 (2022), 371-393.  doi: 10.1007/s12095-021-00530-x. [20] K. Li, L. Qu, B. Sun and C. Li, New results about the boomerang uniformity of permutation polynomials, IEEE Trans. Inform. Theory, 65 (2019), 7542-7553.  doi: 10.1109/TIT.2019.2918531. [21] N. Li, Y. Wu, X. Zeng and X. Tang, On the differential spectrum of a class of power functions over finite fields, arXiv: 2012.04316. [22] Z. Li and H. Yan, Differential spectra of a class of power permutations with niho exponents, Adv. Math. Commun., 2021. doi: 10.3934/amc.2021060. [23] S. Mesnager, B. Mandal and M. Msahli, Survey on recent trends towards generalized differential and boomerang uniformities, Cryptogr. Commun., 14 (2022), 691-735.  doi: 10.1007/s12095-021-00551-6. [24] G. Ness and T. Helleseth, A new family of ternary almost perfect nonlinear mappings, IEEE Trans. Inform. Theory, 53 (2007), 2581-2586.  doi: 10.1109/TIT.2007.899508. [25] K. Nyberg, Differentially uniform mappings for cryptography, Advances in Cryptology—EUROCRYPT '93 (Lofthus, 1993), Lecture Notes in Comput. Sci., 765 (1994), 55–64. doi: 10.1007/3-540-48285-7_6. [26] D. Wagner, The boomerang attack, Fast Software Encryption, Lecture Notes in Comput. Sci., 1636 (1999), 156-170.  doi: 10.1007/3-540-48519-8_12. [27] Y. Xia, X. Zhang, C. Li and T. Helleseth, The differential spectrum of a ternary power mapping, Finite Fields Appl., 64 (2020), 101660, 16 pp. doi: 10.1016/j.ffa.2020.101660. [28] M. Xiong and H. Yan, A note on the differential spectrum of a 4-uniform power function, Finite Fields Appl., 48 (2017), 117-125.  doi: 10.1016/j.ffa.2017.07.008. [29] M. Xiong, H. Yan and P. Yuan, On a conjecture of differentially 8-uniform power function, Des. Codes Cryptogr., 86 (2018), 1601-1621.  doi: 10.1007/s10623-017-0416-7. [30] H. Yan and C. Li, Differential spectra of a class of power permutations with characteristic 5, Des. Codes Cryptogr., 89 (2021), 1181-1191.  doi: 10.1007/s10623-021-00865-9. [31] H. Yan, Y. Xia, C. Li, T. Helleseth, M. Xiong and J. Luo, The differential spectrum of the power mapping $x^{p^n-3}$, IEEE Trans. Inform. Theory. [32] H. Yan, Z. Zhou, J. Weng, J. Wen, T. Helleseth and Q. Wang, Differential spectrum of Kasami power permutations over odd characteristic finite fields, IEEE Trans. Inform. Theory, 65 (2019), 6819-6826.  doi: 10.1109/TIT.2019.2910070. [33] Z. Zha and L. Hu, The boomerang uniformity of power permutations $x^{2^k-1}$ over $F_{2^n}$, Ninth International Workshop on Signal Design and its Applications in Communications (IWSDA), (2019). doi: 10.1109/IWSDA46143.2019.8966114. [34] Z. Zha, G. Kyureghyan and X. Wang, Perfect nonlinear binomials and their semifields, Finite Fields Appl., 15 (2009), 125-133.  doi: 10.1016/j.ffa.2008.09.002. [35] Z. Zha and X. Wang, Almost perfect nonlinear power functions in odd characteristic, IEEE Trans. Inform. Theory, 57 (2011), 4826-4832.  doi: 10.1109/TIT.2011.2145130.
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