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

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<th>Reference</th>
 
<th>Reference</th>
 
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<td>Gold</td>
 
<td>Gold</td>
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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>Kaisa Nyberg, ''Differentially uniform mappings for cryptography'', Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993</ref>
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<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>
 
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  </td>
 
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<td>Kasami</td>
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<td><math>2^{2i} - 2^i + 1</math></td>
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<td><math>\gcd(i,n) = 1</math></td>
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<td><math>i + 1</math></td>
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<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>
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<td>Welch</td>
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<td><math>2^t + 3</math></td>
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<td><math>n = 2t + 1</math></td>
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<td><math>3</math></td>
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<td><ref>Hans Dobbertin, ''Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Welch case'', IEEE Transactions on Information Theory, 45(4):1271-1275, 1999</ref></td>
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<td rowspan="2">Niho</td>
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<td><math>2^t + 2^{t/2} - 1, t</math> even</td>
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<td rowspan="2"><math>n = 2t + 1</math></td>
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<td><math>(t+2)/2</math></td>
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<td rowspan="2"><ref>Hans Dobbertin, ''Almost perfect nonlinear power functions on <math>GF(2^n)</math>: the Niho case'', Information and Computation, 151(1-2):57-72, 1999</ref></td>
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</tr>
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<tr>
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<td><math>2^t + 2^{(3t+1)/2} - 1, t</math> odd</td>
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<td><math>t + 1</math></td>
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<tr>
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<td>Inverse</td>
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<td><math>2^{2t} - 1</math></td>
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<td><math>n = 2t + 1</math></td>
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<td><math>n-1</math></td>
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<td><ref>Thomas Beth and Cunsheng Ding, ''On almost perfect nonlinear permutations'', Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993</ref><ref name="kaisa_ref" />
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</tr>
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<tr>
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<td>Dobbertin</td>
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<td><math>2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1</math></td>
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<td><math>n = 5i</math></td>
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<td><math>i + 3</math></td>
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<td><ref>Hans Dobbertin, ''Almost perfect nonlinear power functions over <math>GF(2^n)</math>: a new case for <math>n</math> divisible by 5'', Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121</ref></td>
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</table>
 
</table>

Revision as of 01:17, 26 November 2018

Family Exponent Conditions Reference
Gold 2 [1][2]
Kasami [3][4]
Welch [5]
Niho even [6]
odd
Inverse [7][2]
Dobbertin [8]
  1. Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968
  2. 2.0 2.1 Kaisa Nyberg, Differentially uniform mappings for cryptography, Workshop on the Theory and Application of Cryptographic Techniques, pp. 55-64, Springer, 1993
  3. Heeralal Janwa and Richard M Wilson, Hyperplane sections of fermat varieties in 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
  4. 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
  5. Hans Dobbertin, Almost perfect nonlinear power functions on : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
  6. Hans Dobbertin, Almost perfect nonlinear power functions on : 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 : a new case for divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121