Difference between revisions of "Some APN functions CCZ-equivalent to x^3 + tr n(x^9) and CCZ-inequivalent to the Gold functions over GF(2^n)"

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Some APN functions CCZ-equivalent to <math>x^3+tr_{n}(x^9)</math> and CCZ-inequivalent to the Gold functions over <math>\mathbb{F}_{2^n}</math><ref>L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001</ref>.
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Some APN functions CCZ-equivalent to <math>x^3+{\mathrm Tr}_{n}(x^9)</math> and CCZ-inequivalent to the Gold functions over <math>\mathbb{F}_{2^n}</math><ref>L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001</ref>.
  
 
<table>
 
<table>
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<tr>
 
<tr>
 
<td><math>1</math></td>
 
<td><math>1</math></td>
<td><math>x^3+tr_n(x^9)+(x^2+x)tr_n(x^3+x^9)</math></td>
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<td><math>x^3+{\mathrm Tr}_n(x^9)+(x^2+x){\mathrm Tr}_n(x^3+x^9)</math></td>
 
<td><math>n\geqslant5</math> odd, <math>\gcd(i,n)=1</math></td>
 
<td><math>n\geqslant5</math> odd, <math>\gcd(i,n)=1</math></td>
 
<td><math>3</math></td>
 
<td><math>3</math></td>
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<tr>
 
<tr>
 
<td><math>2</math></td>
 
<td><math>2</math></td>
<td><math>x^3+tr_n(x^9)+(x^2+x+1)tr_n(x^3)</math></td>
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<td><math>x^3+{\mathrm Tr}_n(x^9)+(x^2+x+1){\mathrm Tr}_n(x^3)</math></td>
 
<td><math>n\geqslant4</math> even, <math>\gcd(i,n)=1</math></td>
 
<td><math>n\geqslant4</math> even, <math>\gcd(i,n)=1</math></td>
 
<td><math>3</math></td>
 
<td><math>3</math></td>
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<tr>
 
<tr>
 
<td><math>3</math></td>
 
<td><math>3</math></td>
<td><math>\Big(x+tr_n^3(x^6+x^{12})+tr_n(x)tr_n^3(x^3+x^{12})\Big)^3+</math> <math>tr_n\Big(\left(x+tr_n^3(x^6+x^{12})+tr_n(x)tr_n^3(x^3+x^{12})\right)^9\Big)</math></td>
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<td><math>\Big(x+{\mathrm Tr}_n^3(x^6+x^{12})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^3+x^{12})\Big)^3+</math> <math>{\mathrm Tr}_n\Big(\left(x+{\mathrm Tr}_n^3(x^6+x^{12})+{\mathrm Tr}_n(x){\mathrm Tr}_n^3(x^3+x^{12})\right)^9\Big)</math></td>
 
<td><math>6|n</math>, <math>\gcd(i,n)=1</math></td>
 
<td><math>6|n</math>, <math>\gcd(i,n)=1</math></td>
 
<td><math>4</math></td>
 
<td><math>4</math></td>
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<tr>
 
<tr>
 
<td><math>4</math></td>
 
<td><math>4</math></td>
<td><math>\left(x^{\frac{1}{3}}+tr_n^3(x+x^4)\right)^{-1}+tr_n\left(\left(\left(x^{\frac{1}{3}}+tr_n^3(x+x^4)\right)^{-1}\right)^{9}\right)</math></td>
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<td><math>\left(x^{\frac{1}{3}}+{\mathrm Tr}_n^3(x+x^4)\right)^{-1}+{\mathrm Tr}_n\left(\left(\left(x^{\frac{1}{3}}+{\mathrm Tr}_n^3(x+x^4)\right)^{-1}\right)^{9}\right)</math></td>
 
<td><math>3|n</math>, <math>n</math> odd</td>
 
<td><math>3|n</math>, <math>n</math> odd</td>
 
<td><math>4</math></td>
 
<td><math>4</math></td>

Latest revision as of 22:11, 10 July 2020

Some APN functions CCZ-equivalent to and CCZ-inequivalent to the Gold functions over [1].

Functions Conditions
odd,
even,
,
, odd
  1. L. Budaghyan, C. Carlet, G. Leander. Constructing new APN functions from known ones. Finite Fields and Their Applications, v. 15, issue 2, pp. 150-159, April 2009. https://doi.org/10.1016/j.ffa.2008.10.001