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2018-12-24T19:23:56Z

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← Older revision		Revision as of 19:23, 24 December 2018
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	~~== Some APN functions <math>CCZ-</math>equivalent to gold functions and EA-enequivalent to power functions over <math>\mathbb{F}_{2^n}</math> ==~~

	Some APN functions <math>CCZ-</math>equivalent to gold functions and EA-enequivalent to power functions over <math>\mathbb{F}_{2^n}</math> (constructed in <ref>Budaghyan, Lilya, Claude Carlet, and Alexander Pott. "New classes of almost bent and almost perfect nonlinear polynomials." IEEE Transactions on Information Theory 52.3 (2006): 1141-1152.</ref>
	~~<table>~~
	~~<tr>~~
	~~<th>Functions</th>~~
	~~<th>Conditions</th>~~
	~~<th><math>d^\circ</math></th>~~
	~~</tr>~~

	~~<tr>~~
	~~<td><math>x^{2^i+1}+(x^{2^i}+tr_n(1)+1)tr(x^{2^i+1}+xtr_n(1))</math></td>~~
	~~<td><math>n\geq4,\ gcd(i,n)=1</math></td>~~
	~~<td>3</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td><math>x+tr_{n/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr_n(x)\, tr_{n/3}(x^{2^i+1}+x^{2^{2i}(2^i+1)})^{2^i+1}</math></td>~~
	~~<td><math>6\|n,\ gcd(i,n)=1</math></td>~~
	~~<td>4</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td><math>x^{2^i+1}+tr_{n/m}(x^{2^i+1})+x^{2^i}tr_{n/m}(x)+xtr_{n/m}(x)^{2^i}+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})~~
	~~+tr_{n/m}(x)]^{\frac{1}{2^i+1}}(x^{2^i}+tr_{n/m}(x)^{2^i}+1)+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})~~
	~~+tr_{n/m}(x)]^{\frac{2^i}{2^i+1}}(x+tr_{n/m}(x))</math></td>~~
	~~<td><math>m\neq n,\ n\ odd,\ m\|n,\ gcd(i,n)=1</math></td>~~
	~~<td><math>m+2</math></td>~~
	~~</tr>~~

	~~</table>~~

Fady: /* Some APN functions CCZ-equivalent to gold functions and EA-enequivalent to power functions over \mathbb{F}_{2^n} */ new section

2018-12-24T19:08:44Z

Some APN functions CCZ-equivalent to gold functions and EA-enequivalent to power functions over \mathbb{F}_{2^n}: new section

New page

== Some APN functions <math>CCZ-</math>equivalent to gold functions and EA-enequivalent to power functions over <math>\mathbb{F}_{2^n}</math> ==

Some APN functions <math>CCZ-</math>equivalent to gold functions and EA-enequivalent to power functions over <math>\mathbb{F}_{2^n}</math> (constructed in <ref>Budaghyan, Lilya, Claude Carlet, and Alexander Pott. "New classes of almost bent and almost perfect nonlinear polynomials." IEEE Transactions on Information Theory 52.3 (2006): 1141-1152.</ref>
<table>
<tr>
<th>Functions</th>
<th>Conditions</th>
<th><math>d^\circ</math></th>
</tr>

<tr>
<td><math>x^{2^i+1}+(x^{2^i}+tr_n(1)+1)tr(x^{2^i+1}+xtr_n(1))</math></td>
<td><math>n\geq4,\ gcd(i,n)=1</math></td>
<td>3</td>
</tr>

<tr>
<td><math>x+tr_{n/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr_n(x)\, tr_{n/3}(x^{2^i+1}+x^{2^{2i}(2^i+1)})^{2^i+1}</math></td>
<td><math>6|n,\ gcd(i,n)=1</math></td>
<td>4</td>
</tr>

<tr>
<td><math>x^{2^i+1}+tr_{n/m}(x^{2^i+1})+x^{2^i}tr_{n/m}(x)+xtr_{n/m}(x)^{2^i}+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})
+tr_{n/m}(x)]^{\frac{1}{2^i+1}}(x^{2^i}+tr_{n/m}(x)^{2^i}+1)+[tr_{n/m}(x)^{2^i+1}+tr_{n/m}(x^{2^i+1})
+tr_{n/m}(x)]^{\frac{2^i}{2^i+1}}(x+tr_{n/m}(x))</math></td>
<td><math>m\neq n,\ n\ odd,\ m|n,\ gcd(i,n)=1</math></td>
<td><math>m+2</math></td>
</tr>

</table>

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Fady: /* Some APN functions CCZ-equivalent to gold functions and EA-enequivalent to power functions over \mathbb{F}_{2^n} */ new section