Difference between revisions of "Known infinite families of quadratic APN polynomials over GF(2^n)"

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<td>C11</td>
 
<td>C11</td>
<td><math>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></td>
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<td><math>L(x)^{2^i}x+L(x)x^{2^i}</math></td>
<td><math>n=3m, m \ \text{odd}, L(x)=ax^{2^{2m}}+bx^{2^{m}}+cx</math> satisfies the conditions in Lemma 8 of [7]</td>
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<td><math>n=km, \gcd(n,i)=1, L(x)=\sum_{j=0}^{k-1}a_jx^{2^{jm}}</math> satisfies the conditions in Theorem 3.6 of [7]</td>
<td><ref>Villa I, Budaghyan L, Calderini M, Carlet C, Coulter R. On Isotopic Construction of APN Functions. SETA 2018</ref></td>
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<td><ref>Villa I, Budaghyan L, Calderini M, Carlet C, Coulter R. Constructing APN functions through isotopic shift. Cryptology ePrint Archive, Report 2018/769</ref></td>
 
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<td><math>n = 2m, m odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}, i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math></td>
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<td><math>n = 2m, m\ odd, 3 \nmid m, (a,b,c) = (\beta, \beta^2, 1), \beta \text{ primitive in } \mathbb{F}_{2^2}, i \in \{ m-2, m, 2m-1, (m-2)^{-1} \mod n \}</math></td>
 
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Revision as of 09:53, 30 October 2019

Functions Conditions References
C1-C2 [1]
C3 , s.t. [2]
C4 [3]
C5 , [4]
C6 [4]
C7-C9 [5]
C10 even, and even, [6]
C11 satisfies the conditions in Theorem 3.6 of [7] [7]
C12 [8]
  1. Budaghyan L, Carlet C, Leander G. Two classes of quadratic APN binomials inequivalent to power functions. IEEE Transactions on Information Theory. 2008 Sep;54(9):4218-29.
  2. Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.
  3. 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.
  4. 4.0 4.1 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.
  5. Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.
  6. Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.
  7. Villa I, Budaghyan L, Calderini M, Carlet C, Coulter R. Constructing APN functions through isotopic shift. Cryptology ePrint Archive, Report 2018/769
  8. Budaghyan L, Helleseth T, Kaleyski N. A new family of APN quadrinomials. Cryptology ePrint Archive, Report 2019/994