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

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<td><math>2^i + 1</math></td>
 
<td><math>2^i + 1</math></td>
 
<td><math>\gcd(i,n) = 1</math></td>
 
<td><math>\gcd(i,n) = 1</math></td>
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<td>2</td>
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<td><math>2</math></td>
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<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> <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).</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>
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<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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<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).</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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<td><math>n = 2t + 1</math></td>
 
<td><math>n = 2t + 1</math></td>
 
<td><math>n-1</math></td>
 
<td><math>n-1</math></td>
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<td><ref name="kaisa_ref" />Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76). Springer, Berlin, Heidelberg.</ref>
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<td><ref name="kaisa_ref" /><ref>Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76). </ref>
 
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<td><math>n = 5i</math></td>
 
<td><math>n = 5i</math></td>
 
<td><math>i + 3</math></td>
 
<td><math>i + 3</math></td>
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<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121). Springer, Berlin, Heidelberg.</ref></td>
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<td><ref>Dobbertin H. Almost perfect nonlinear power functions on <math>GF(2^n)</math>: a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).</ref></td>
 
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Revision as of 12:56, 11 February 2019

The following table provides a summary of all known infinite families of power APN functions of the form .

Family Exponent Conditions Reference
Gold [1][2]
Kasami [3][4]
Welch [5]
Niho even [6]
odd
Inverse [2][7]
Dobbertin [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).
  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).
  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. ↑ Dobbertin H. Almost perfect nonlinear power functions on : the Welch case. IEEE Transactions on Information Theory. 1999 May;45(4):1271-5.
  6. ↑ Dobbertin H. Almost perfect nonlinear power functions on : the Niho case. Information and Computation. 1999 May 25;151(1-2):57-72.
  7. ↑ Beth T, Ding C. On almost perfect nonlinear permutations. InWorkshop on the Theory and Application of of Cryptographic Techniques 1993 May 23 (pp. 65-76).
  8. ↑ Dobbertin H. Almost perfect nonlinear power functions on : a new case for n divisible by 5. InFinite Fields and Applications 2001 (pp. 113-121).
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