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

Revision as of 14:07, 15 January 2019

The following table provides a summary of all known infinite families of power APN functions of the form $F(x)=x^{d}$ .

Family	Exponent	Conditions	$\deg(x^{d})$	Reference
Gold	$2^{i}+1$	$\gcd(i,n)=1$	2	^[1]^[2]
Kasami	$2^{2i}-2^{i}+1$	$\gcd(i,n)=1$	$i+1$	^[3]^[4]
Welch	$2^{t}+3$	$n=2t+1$	$3$	^[5]
Niho	$2^{t}+2^{t/2}-1,t$ even	$n=2t+1$	$(t+2)/2$	^[6]
Niho	$2^{t}+2^{(3t+1)/2}-1,t$ odd	$n=2t+1$	$t+1$	^[6]
Inverse	$2^{2t}-1$	$n=2t+1$	$n-1$	^[2]^[7]
Dobbertin	$2^{4i}+2^{3i}+2^{2i}+2^{i}-1$	$n=5i$	$i+3$	^[8]

↑ Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968
↑ ^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
↑ Heeralal Janwa and Richard M Wilson, Hyperplane sections of Fermat varieties in $P^{3}$ 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
↑ 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
↑ Hans Dobbertin, Almost perfect nonlinear power functions on $GF(2^{n})$ : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999
↑ Hans Dobbertin, Almost perfect nonlinear power functions on $GF(2^{n})$ : the Niho case, Information and Computation, 151(1-2):57-72, 1999
↑ Thomas Beth and Cunsheng Ding, On almost perfect nonlinear permutations, Workshop on the Theory and Application of Cryptographic Techniques, pp. 65-76, Springer, 1993
↑ Hans Dobbertin, Almost perfect nonlinear power functions over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle GF(2^n)} : a new case for $n$ divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121

[1] Robert Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions (corresp.), IEEE transactions on Information Theory, 14(1):154-156, 1968

[kaisa_ref-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 $P^{3}$ 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 $GF(2^{n})$ : the Welch case, IEEE Transactions on Information Theory, 45(4):1271-1275, 1999

[6] Hans Dobbertin, Almost perfect nonlinear power functions on $GF(2^{n})$ : 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle GF(2^n)} : a new case for $n$ divisible by 5, Proceedings of the fifth conference on Finite Fields and Applications FQ5, pp.113-121

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@@ Line 25: / Line 25: @@
 <td><math>\gcd(i,n) = 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>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>
 </tr>
@@ Line 54: / Line 54: @@
 <td><math>n = 2t + 1</math></td>
 <td><math>n-1</math></td>
-<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" />
+<td><ref name="kaisa_ref" /><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>
 </tr>

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

Revision as of 14:07, 15 January 2019

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