Known infinite families of APN power functions over GF(2^n): Difference between revisions

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<td><math>\gcd(i,n) = 1</math></td>
<td><math>\gcd(i,n) = 1</math></td>
<td>2</td>
<td>2</td>
<td> <ref>Robert Gold, ''Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.)'', IEEE transactions on Information Theory, 14(1):154-156, 1968</ref><ref name="kaisa_ref">Kaisa Nyberg, ''Differentially uniform mappings for cryptography'', Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993</ref>
<td> <ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1988;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). Springer, Berlin, Heidelberg.</ref>
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<td><math>\gcd(i,n) = 1</math></td>
<td><math>\gcd(i,n) = 1</math></td>
<td><math>i + 1</math></td>
<td><math>i + 1</math></td>
<td><ref>Heeralal Janwa and Richard M Wilson, ''Hyperplane sections of Fermat varieties in <math>P^3</math> in char. 2 and some applications to cyclic codes'', International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, pp. 180-194, Springer, 1993</ref><ref>Tadao Kasami, ''The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes'', Information and Control, 18(4):369-394, 1971</ref></td>
<td><ref>Janwa H, Wilson RM. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. InInternational Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes 1993 May 10 (pp. 180-194). Springer, Berlin, Heidelberg.</ref><ref>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.</ref></td>
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Revision as of 09:56, 11 February 2019

The following table provides a summary of all known infinite families of power APN functions of the form [math]\displaystyle{ F(x) = x^d }[/math].

Family Exponent Conditions [math]\displaystyle{ \deg(x^d) }[/math] Reference
Gold [math]\displaystyle{ 2^i + 1 }[/math] [math]\displaystyle{ \gcd(i,n) = 1 }[/math] 2 [1][2]
Kasami [math]\displaystyle{ 2^{2i} - 2^i + 1 }[/math] [math]\displaystyle{ \gcd(i,n) = 1 }[/math] [math]\displaystyle{ i + 1 }[/math] [3][4]
Welch [math]\displaystyle{ 2^t + 3 }[/math] [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ 3 }[/math] [5]
Niho [math]\displaystyle{ 2^t + 2^{t/2} - 1, t }[/math] even [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ (t+2)/2 }[/math] [6]
[math]\displaystyle{ 2^t + 2^{(3t+1)/2} - 1, t }[/math] odd [math]\displaystyle{ t + 1 }[/math]
Inverse [math]\displaystyle{ 2^{2t} - 1 }[/math] [math]\displaystyle{ n = 2t + 1 }[/math] [math]\displaystyle{ n-1 }[/math] [2][7]
Dobbertin [math]\displaystyle{ 2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1 }[/math] [math]\displaystyle{ n = 5i }[/math] [math]\displaystyle{ i + 3 }[/math] [8]
  1. ↑ Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inf. Theory. 1988;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). Springer, Berlin, Heidelberg.
  3. ↑ Janwa H, Wilson RM. Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. InInternational Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes 1993 May 10 (pp. 180-194). Springer, Berlin, Heidelberg.
  4. ↑ 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.
  5. ↑ Hans Dobbertin, Almost perfect nonlinear power functions on [math]\displaystyle{ GF(2^n) }[/math]: the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
  6. ↑ Hans Dobbertin, Almost perfect nonlinear power functions on [math]\displaystyle{ GF(2^n) }[/math]: the Niho case, Information and Computation, 151(1-2):57-72, 1999
  7. ↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
  8. ↑ Hans Dobbertin, Almost perfect nonlinear power functions over [math]\displaystyle{ GF(2^n) }[/math]: a new case for [math]\displaystyle{ n }[/math] divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121
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