Constructions of complete permutation polynomials

Based on the Feistel and MISTY structures, this paper presents several new constructions of complete permutation polynomials (CPPs) of the finite field $${\mathbb {F}}_{2^{n}}^2$$F2n2 for a positive integer n and three constructions of CPPs over $${\mathbb {F}}_{p^{n}}^m$$Fpnm for any prime p and positive integer $$m\ge 2$$m≥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.

[1]  Dengguo Feng,et al.  Loiss: A Byte-Oriented Stream Cipher , 2011, IWCC.

[2]  Serge Vaudenay,et al.  On the Lai-Massey Scheme , 1999, ASIACRYPT.

[3]  Lei Hu,et al.  Several classes of complete permutation polynomials , 2014, Finite Fields Their Appl..

[4]  Serge Vaudenay,et al.  On the Need for Multipermutations: Cryptanalysis of MD4 and SAFER , 1994, FSE.

[5]  Dongdai Lin,et al.  On constructing complete permutation polynomials over finite fields of even characteristic , 2013, Discret. Appl. Math..

[6]  Serge Vaudenay,et al.  Black Box Cryptanalysis of Hash Networks Based on Multipermutations , 1994, EUROCRYPT.

[7]  Tor Helleseth,et al.  Some classes of monomial complete permutation polynomials over finite fields of characteristic two , 2014, Finite Fields Their Appl..

[8]  Henry B. Mann,et al.  The Construction of Orthogonal Latin Squares , 1942 .

[9]  Daqing Wan.  On a problem of Niederreiter and Robinson about finite fields , 1986 .

[10]  Smile Markovski,et al.  Generating huge quasigroups from small non-linear bijections via extended Feistel function , 2009 .

[11]  Xiwang Cao,et al.  Complete permutation polynomials over finite fields of odd characteristic , 2013, Finite Fields Their Appl..

[12]  H. Niederreiter,et al.  Complete mappings of finite fields , 1982, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[13]  Tor Helleseth,et al.  Some classes of complete permutation polynomials over $\mathbb{F}_q $ , 2015 .

[14]  Xiang-dong Hou,et al.  Permutation polynomials over finite fields - A survey of recent advances , 2015, Finite Fields Their Appl..

[15]  Smile Markovski,et al.  Quasigroup Representation of Some Feistel and Generalized Feistel Ciphers , 2012, ICT Innovations.

[16]  Lei Hu,et al.  Constructing permutations and complete permutations over finite fields via subfield-valued polynomials , 2015, Finite Fields Their Appl..

[17]  Sugata Gangopadhyay,et al.  Investigations on Bent and Negabent Functions via the Nega-Hadamard Transform , 2012, IEEE Transactions on Information Theory.

[18]  Smile Markovski,et al.  Shapeless Quasigroups Derived by Feistel Orthomorphisms , 2012 .

[19]  Xingju Cai,et al.  A proximal alternating linearization method for minimizing the sum of two convex functions , 2015 .

[20]  Daniel Panario,et al.  Handbook of Finite Fields , 2013, Discrete mathematics and its applications.

[21]  Sumanta Sarkar,et al.  On Some Permutation Binomials of the Form $x^{\frac{2^n-1}{k}+1} +ax$ over $\mathbb{F}_{2^n}$ : Existence and Count , 2012, WAIFI.

[22]  Enes Pasalic,et al.  A note on complete polynomials over finite fields and their applications in cryptography , 2014, Finite Fields Their Appl..

[23]  Daniele Bartoli,et al.  On monomial complete permutation polynomials , 2016, Finite Fields Their Appl..

[24]  Mitsuru Matsui,et al.  New Block Encryption Algorithm MISTY , 1997, FSE.

[25]  V. A. Zinoviev,et al.  Permutation and complete permutation polynomials , 2015, Finite Fields Their Appl..

[26]  Cunsheng Ding,et al.  Permutation polynomials over finite fields from a powerful lemma , 2011, Finite Fields Their Appl..

[27]  L. Mittenthal,et al.  Block Substitutions Using Orthomorphic Mappings , 1995 .