Differentially 4-uniform permutation

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Functions Conditions References
[math]\displaystyle{ x^{2^i+1} }[/math] [math]\displaystyle{ gcd(i,n) = 2, n = 2t }[/math] and t is odd [1][2]
[math]\displaystyle{ x^{2^{2i}-2^i+1} }[/math] [math]\displaystyle{ gcd(i,n) = 2, n = 2t }[/math] and t is odd [3]
[math]\displaystyle{ x^{2^n-2} }[/math] [math]\displaystyle{ n = 2t }[/math] (inverse) [2][4]
[math]\displaystyle{ x^{2^{2t}-2^t+1} }[/math] [math]\displaystyle{ n = 4t }[/math] and t is odd [5]
[math]\displaystyle{ \alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}} }[/math] [math]\displaystyle{ n = 3t, t/2 }[/math] is odd, [math]\displaystyle{ gcd(n,s) = 2, 3|t + s }[/math] and [math]\displaystyle{ \alpha }[/math] is a primitive element in [math]\displaystyle{ \mathbb{F}_{2^n} }[/math] [6]
[math]\displaystyle{ x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1}) }[/math] [math]\displaystyle{ n=2t }[/math] is even [7]
[math]\displaystyle{ x^{-1} + \mathrm {Tr}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)}) }[/math] [math]\displaystyle{ n=2t }[/math] and [math]\displaystyle{ 2\leq k \leq t-1 }[/math] [7]
[math]\displaystyle{ a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3 }[/math] [math]\displaystyle{ n=3m, m \ \text{odd}, L(x)=ax^{2^{2m}}+bx^{2^{m}}+cx }[/math] satisfies the conditions in Lemma 8 of [7] [8]
  1. Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.
  2. 2.0 2.1 Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).
  3. Kasami T. The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control. 1971 May 1;18(4):369-94.
  4. Lachaud G, Wolfmann J. The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE transactions on information theory. 1990 May;36(3):686-92.
  5. Bracken C, Leander G. A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields and Their Applications. 2010 Jul 1;16(4):231-42.
  6. Bracken C, Tan CH, Tan Y. Binomial differentially 4 uniform permutations with high nonlinearity. Finite Fields and Their Applications. 2012 May 1;18(3):537-46.
  7. 7.0 7.1 >Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over [math]\displaystyle{ {\mathbb F} _ {2^{2k}} }[/math]. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.
  8. Villa I, Budaghyan L, Calderini M, Carlet C, & Coulter R. On Isotopic Construction of APN Functions. SETA 2018