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Constructions of complete permutation polynomials. (English) Zbl 1398.05012
Summary: Based on the Feistel and MISTY structures, this paper presents several new constructions of complete permutation polynomials (CPPs) of the finite field \({\mathbb {F}}_{2^{n}}^2\) for a positive integer \(n\) and three constructions of CPPs over \({\mathbb {F}}_{p^{n}}^m\) for any prime \(p\) and positive integer \(m\geq 2\). In addition, we investigate the upper bound on the algebraic degree of these CPPs and show that some of them can have nearly optimal algebraic degree.

MSC:
05A05 Permutations, words, matrices
11T06 Polynomials over finite fields
11T55 Arithmetic theory of polynomial rings over finite fields
Software:
MISTY
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References:
[1] Bartolia, D.; Giulietti, M.; Zinib, G., On monomial complete permutation polynomials, Finite Fields Appl., 41, 132-158, (2016) · Zbl 1372.11107
[2] Bassalygo, L.; Zinoviev, V., Permutation and complete permutation polynomials, Finite Fields Appl., 33, 198-211, (2015) · Zbl 1368.11126
[3] Diffie W., Ledin G. (translators): SMS4 encryption algorithm for wireless networks. https://eprint.iacr.org/2008/329.pdf.
[4] Feng, Dengguo; Feng, Xiutao; Zhang, Wentao; Fan, Xiubin; Wu, Chuankun, Loiss: A Byte-Oriented Stream Cipher, 109-125, (2011), Berlin, Heidelberg · Zbl 1272.94029
[5] Hou, X., Permutation polynomials over finite fields-a survey of recent advances, Finite Fields Appl., 32, 82-119, (2015) · Zbl 1325.11128
[6] Lidl R., Niederreiter H.: Finite Fields Encycl. Math. Appl. Cambridge University Press, Cambridge (1997).
[7] Mann, HB, The construction of orthogonal Latin squares, Ann. Math. Stat., 13, 418-423, (1942) · Zbl 0060.02706
[8] Markovski, S.; Mileva, A., Generating huge quasigroups from small non-linear bijections via extended Feistel function, Quasigroups Relat. Syst., 17, 91-106, (2009) · Zbl 1175.20058
[9] Matsui M.: New block encryption algorithm MISTY. In: Fast Software Encryption—FSE’97. Lect. Notes Comput. Sci, vol. 1267, pp. 54-68. Springer, New York (1997) · Zbl 1385.94061
[10] Mileva, Aleksandra; Markovski, Smile, Quasigroup Representation of Some Feistel and Generalized Feistel Ciphers, 161-171, (2013), Berlin, Heidelberg · Zbl 1315.94093
[11] Mileva, A.; Markovski, S., Shapeless quasigroups derived by Feistel orthomorphisms, Glas. Mat., 47, 333-349, (2012) · Zbl 1269.94027
[12] Mittenthal L.: Nonlinear dynamic substitution devices and methods for block substitutions employing coset decompositions and direct geometric generation. US Patent 5647001 (1997).
[13] Mittenthal, L., Block substitutions using orthomorphic mappings, Adv. Appl. Math., 16, 59-71, (1995) · Zbl 0863.20012
[14] Mullen G.L., Panario D.: Handbook of Finite Fields. Taylor-Francis, Boca Raton (2013). · Zbl 1319.11001
[15] Muratovic-Ribic, A.; Pasalic, E., A note on complete mapping polynomials over finite fields and their applications in cryptography, Finite Fields Appl., 25, 306-315, (2014) · Zbl 1302.11096
[16] National Institute of Standards and Technology.: Data Encryption Standard, FIPS Publication 46-2 (1993). http://www.itl.nist.gov/fipspubs/fip46-2.htm.
[17] Niederreiter, H.; Robinson, KH, Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 197-212, (1982) · Zbl 0495.12018
[18] Sarkar S., Bhattacharya S., Cesmelioglu A.: On some permutation binomials of the form \(x^{\frac{2^n-1}{k}+1}+ax\) over \({\mathbb{F}}_{2^n}\): existence and count. In: WAIFI, 2012. Lect. Notes Comput. Sci., vol. 7369, pp. 236-246. Springer, New York (2012). · Zbl 1292.11132
[19] Schnorr, C. P.; Vaudenay, S., Black box cryptanalysis of hash networks based on multipermutations, 47-57, (1995), Berlin, Heidelberg · Zbl 0909.94013
[20] Specification of SMS4, block cipher for WLAN products-SMS4 (in Chinese). http://www.oscca.gov.cn/UpFile/200621016423197990.pdf.
[21] Stǎnicǎ, P.; Gangopadhyay, S.; Chaturvedi, A.; Gangopadhyay, AK; Maitra, S., Investigations on bent and negabent functions via the negaHadamard transform, IEEE Trans. Inf. Theory, 58, 4064-4072, (2012) · Zbl 1365.94684
[22] Tu, Z.; Zeng, X.; Hu, L., Several classes of complete permutation polynomials, Finite Fields Appl., 25, 182-193, (2014) · Zbl 1284.05012
[23] Tuxanidy A., Wang Q.: Compositional inverses, complete mappings, orthogonal Latin squares and bent functions. arXiv:1409.6540 [math.NT] (2014).
[24] Vaudenay S.: On the Lai-Massey scheme. In: Advances in Cryptology—ASIACRYPT’99. Lect. Notes Comput. Sci., vol. 1716, pp. 8-19. Springer, New York (1999). · Zbl 0977.94044
[25] Vaudenay S.: On the need for multipermutations: cryptanalysis of MD4 and SAFER. In: Fast Software Encryption—FSE’94. Lect. Notes Comput. Sci., vol. 1008, pp. 286-297. Springer, New York (1994). · Zbl 0939.94542
[26] Wan, D., On a problem of Niederreiter and Robinson about finite fields, J. Aust. Math. Soc. A, 41, 336-338, (1986) · Zbl 0607.12009
[27] Wu, B.; Lin, D., On constructing complete permutation polynomials over finite fields of even characteristic, Discret. Appl. Math., 184, 213-222, (2015) · Zbl 1311.05009
[28] Wu, G.; Li, N.; Helleseth, T.; Zhang, Y., Some classes of monomial complete permutation polynomials over finite fields of characteristic two, Finite Fields Appl., 28, 148-165, (2014) · Zbl 1314.11073
[29] Wu, G.; Li, N.; Helleseth, T.; Zhang, Y., More classes of complete permutation polynomials over \({\mathbb{F}}_{q}\), Sci. China Math., 58, 1-14, (2015) · Zbl 1325.05013
[30] Xu, G.; Cao, X., Complete permutation polynomials over finite fields of odd characteristic, Finite Fields Appl., 31, 228-240, (2015) · Zbl 1320.11121
[31] Yuan Y., Tong Y., Zhang H.: Complete mapping polynomials over finite field \({\mathbb{F}}_{16}\). In: Arithmetic of Finite Fields. Lect. Notes Comput. Sci., vol. 4547, pp. 147-158. Springer, New York (2007). · Zbl 1213.11193
[32] Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 560-574, (2011) · Zbl 1258.11100
[33] Zha, Z.; Hu, L.; Cao, X., Constructing permutations and complete permutations over finite fields via subfield-valued polynomials, Finite Fields Appl., 31, 162-177, (2015) · Zbl 1320.11123
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