Differentially 4-uniform permutation: Difference between revisions

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<td><math>x^{2^i+1}</math></td>
<td><math>x^{2^i+1}</math></td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref></td>
<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref><ref name="kaisa_ref">Nyberg K. Differentially uniform mappings for cryptography. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 55-64).</ref></td>
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<td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td>
<td><math>x^{2^n-2}</math></td>
<td><math>a\neq 0</math></td>
<td><math> n = 2t</math> (inverse)</td>
<td><ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref></td>
<td><ref name="kaisa_ref" /><ref>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.</ref></td>
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<td><math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math></td>
<td><math>x^{2^{2t}-2^t+1}</math></td>
<td><math>3|n </math>, <math>a\ne0</math></td>
<td><math>n = 4t</math> and t is odd</td>
<td><ref name="2_ref">Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. InInformation Theory Workshop, 2009. ITW 2009. IEEE 2009 Oct 11 (pp. 374-378). IEEE.</ref></td>
<td><ref>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.</ref></td>
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<td><math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td><math>\alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}}</math></td>
<td> <math>3|n, a \ne 0</math></td>
<td> <math>n = 3t, t/2</math> is odd, <math>gcd(n,s) = 2, 3|t + s</math> and <math>\alpha</math> is a primitive element in <math>\mathbb{F}_{2^n}</math></td>
<td><ref name="2_ref" /></td>
<td><ref>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.</ref></td>
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<td><math>ux^{2^s+1}+u^{2^k} x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^k+1}x^{2^{s}+2^{k+s}}</math></td>
<td><math>x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1})</math></td>
<td><math>n=3k, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* </math></td>
<td><math>n=2t</math> is even</td>
<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.</ref></td>
<td><ref name="kai_ref">>Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over <math>{\mathbb F} _ {2^{2k}} </math>. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.</ref></td>
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<td><math>(x+x^{2{^m}})^{2^k+1}+u'(ux+u^{2^{m}} x^{2^{m}})^{(2^k+1)2^i}+u(x+x^{2^{m}})(ux+u^{2^{m}} x^{2^{m}})</math></td>
<td><math>x^{-1} + \mathrm {Tr}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)})</math></td>
<td><math>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1</math> and <math> i \geqslant 2</math> even, <math>u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }</math></td>
<td><math>n=2t</math> and <math>2\leq k \leq t-1</math></td>
<td><ref>Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.</ref></td>
<td><ref name="kai_ref" /></td>
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Revision as of 10:36, 13 June 2019

Functions Conditions References
x2i+1 gcd(i,n)=2,n=2t and t is odd [1][2]
x22i−2i+1 gcd(i,n)=2,n=2t and t is odd [3]
x2n−2 n=2t (inverse) [2][4]
x22t−2t+1 n=4t and t is odd [5]
αx2s+1+α2tx2−t+2t+s n=3t,t/2 is odd, gcd(n,s)=2,3|t+s and α is a primitive element in 𝔽2n [6]
x−1+Tr(x+(x−1+1)−1) n=2t is even [7]
x−1+Tr(x−3(2k+1)+(x−1+1)3(2k+1)) n=2t and 2≤k≤t−1 [7]
a2x22m+1+1+b2x2m+1+1+ax22m+2+bx2m+2+(c2+c)x3 n=3m,m odd,L(x)=ax22m+bx2m+cx 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 𝔽22k. 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
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