Differentially 4-uniform permutation - Revision history

Fady at 13:27, 13 June 2019

2019-06-13T13:27:39Z

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			<td><math>x^{-1} + t(x^{2^{s}}+x)^{2^{sn}-1}</math></td>
			<td> <math> s</math> is even <math> t \in {\mathbb F}^{} _ {2^{s}},</math> or <math>s, n</math> are odd, <math>t \in {\mathbb F}^{} _ {2^{s}}</math> </td>
			<td><ref>Zha Z, Hu L, Sun S. Constructing new differentially 4-uniform permutations from the inverse function. Finite Fields and Their Applications. 2014 Jan 1;25:64-78.</ref></td>
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			<td><math>x^{2^{k}+1} + t(x^{2^{s}}+x)^{2^{sn}-1}</math></td>
			<td><math>n, s</math> are odd, <math> t \in {\mathbb F}^{*} _ {2^{s}}, gcd(k, sn) = 1 </math></td>
			<td><ref>Xu G, Cao X, Xu S. Constructing new differentially 4-uniform permutations and APN functions over finite fields. Cryptography and Communications-Discrete Structures, Boolean Functions and Sequences. Pre-print. 2014.</ref></td>
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			<td><math>(x, x_n) \to</math>
			<math>((1+x_{n})x^{-1}+x_{n}\alpha x^{-1}, f(x, x_{n}))</math>
			<td> <math>n</math> is even <math> x, \alpha \in {\mathbb F} _ {2^{n-1}}, x_n \in {\mathbb F} _ {2}, \mathrm{Tr}^{n-1}_1(\alpha) = \mathrm{Tr}^{n-1}_1(\frac{1}{\alpha}) = 1,</math>
			<math> f(x, x_n)</math> is <math>(n, 1)-</math>function

			<td><ref>Carlet C, Tang D, Tang X, Liao Q. New construction of differentially 4-uniform bijections. InInternational Conference on Information Security and Cryptology 2013 Nov 27 (pp. 22-38). Springer, Cham.</ref></td>
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Fady at 11:18, 13 June 2019

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			<td><math>\displaystyle\sum_{i=0}^{2^{n}-3} x^{i}</math></td>
			<td><math>n=2t, </math> t is odd</td>
			<td><ref>Yu Y, Wang M, Li Y. Constructing low differential uniformity functions from known ones. Chinese Journal of Electronics. 2013;22(3):495-9.</ref></td>
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Fady at 11:09, 13 June 2019

2019-06-13T11:09:45Z

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	<td><math>x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1})</math></td>		<td><math>x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1})</math></td>
	<td><math>n=2t</math> is even</td>		<td><math>n=2t</math> is even</td>
	<td><ref name="kai_ref">>Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over <math>{\mathbb F} _ {2^{2k}} </math>. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.</ref></td>		<td><ref name="kai_ref">Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over <math>{\mathbb F} _ {2^{2k}} </math>. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.</ref></td>
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	<td><ref>Li Y, Wang M. Constructing differentially 4-uniform permutations over<math>{\mathbb F} _ {2^{2m}} </math> from quadratic APN permutations over <math>{\mathbb F} _ {2^{2m+1}}</math>. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.</ref></td>		<td><ref>Li Y, Wang M. Constructing differentially 4-uniform permutations over<math>{\mathbb F} _ {2^{2m}} </math> from quadratic APN permutations over <math>{\mathbb F} _ {2^{2m+1}}</math>. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.</ref></td>
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Fady at 11:04, 13 June 2019

2019-06-13T11:04:32Z

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	<td><math>~~a^2x~~^{2^{~~2m+1~~}+1}~~+b^2x^~~{2^{m+1}+1}~~+ax^~~{2^{2m}+2}~~+bx^~~{2^{m}+2}~~+(c^2+c)x^3~~</math></td>		<td><math>L_u(F^{-1}(x))\|_{H_u}</math></td>
	<td><math>~~n=3m, m~~ \ ~~\text{odd~~}~~, L(x)=ax^~~{2^{2m}}~~+bx^~~{2^{m}}~~+cx~~</math> ~~satisfies the conditions in Lemma 8 of [7]</td>~~		<td><math>n=2t,F(x)</math> is a quadratic APN permutation on <math>{\mathbb F} _ {2^{n+1}}, u \in {\mathbb F^{*}} _ {2^{n+1}}</math></td>
	~~<td><ref>Villa I~~, ~~Budaghyan L, Calderini M, Carlet C, & Coulter R~~. ~~On Isotopic Construction of APN Functions~~. ~~SETA 2018~~</ref></td>		<td><ref>Li Y, Wang M. Constructing differentially 4-uniform permutations over<math>{\mathbb F} _ {2^{2m}} </math> from quadratic APN permutations over <math>{\mathbb F} _ {2^{2m+1}}</math>. Designs, Codes and Cryptography. 2014 Aug 1;72(2):249-64.</ref></td>
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Fady at 10:36, 13 June 2019

2019-06-13T10:36:17Z

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	<td><math>x^{2^i+1}</math></td>		<td><math>x^{2^i+1}</math></td>
	<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>		<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
	<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref></td>		<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;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></td>
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	<td><math>x^~~3+a~~^{-1} ~~\mathrm {Tr}_n (a^3x^9)~~</math></td>		<td><math>x^{2^n-2}</math></td>
	<td><math>~~a\neq 0~~</math></td>		<td><math> n = 2t</math> (inverse)</td>
	<td><ref>~~Budaghyan L~~, ~~Carlet C, Leander G~~. ~~Constructing new APN functions from known ones~~. ~~Finite Fields and Their Applications~~. ~~2009 Apr 1~~;15(2):~~150~~-9.</ref></td>		<td><ref name="kaisa_ref" /><ref>Lachaud G, Wolfmann J. The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE transactions on information theory. 1990 May;36(3):686-92.</ref></td>
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	<td><math>x^~~3+a~~^{-~~1} \mathrm {Tr}_n^3 (a~~^~~3x^9~~+~~a^6x^{18~~})</math></td>		<td><math>x^{2^{2t}-2^t+1}</math></td>
	<td><math>3\|n </math>~~, <math>a\ne0</math>~~</td>		<td><math>n = 4t</math> and t is odd</td>
	<td><ref ~~name="2_ref"~~>~~Budaghyan L, Carlet~~ C, Leander G. ~~On a construction~~ of ~~quadratic APN functions. InInformation Theory Workshop, 2009~~. ~~ITW 2009~~. ~~IEEE 2009 Oct 11~~ (~~pp. 374~~-~~378). IEEE~~.</ref></td>		<td><ref>Bracken C, Leander G. A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields and Their Applications. 2010 Jul 1;16(4):231-42.</ref></td>
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	<td><math>x^3+~~a^{-~~1} \~~mathrm~~{Tr}~~_n^3(a^6x~~^{18}+a^{12}~~x^{36~~})</math></td>		<td><math>\alpha x^{2^s+1}+\alpha^{2^t}x^{{2-t}+2^{t+s}}</math></td>
	<td> <math>3\|n, a \~~ne 0~~</math></td>		<td> <math>n = 3t, t/2</math> is odd, <math>gcd(n,s) = 2, 3\|t + s</math> and <math>\alpha</math> is a primitive element in <math>\mathbb{F}_{2^n}</math></td>
	<td><ref ~~name="2_ref"~~ /></td>		<td><ref>Bracken C, Tan CH, Tan Y. Binomial differentially 4 uniform permutations with high nonlinearity. Finite Fields and Their Applications. 2012 May 1;18(3):537-46.</ref></td>
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	<td><math>ux^{~~2^s+~~1}+u^{~~2^k~~} x~~^{2^{-k}~~+~~2^{k+s}}+vx^{2~~^{-k}+1~~}+wu~~^{~~2^k+~~1}~~x^{2^{s}+2^{k+s}}~~</math></td>		<td><math>x^{-1} + \mathrm {Tr}(x+ (x^{-1}+1)^{-1})</math></td>
	<td><math>n=~~3k, \gcd(k,3)~~=~~\gcd(s~~,~~3k)=1~~, v, ~~w\in~~\mathbb{F}_{2^~~k}, vw \ne 1, 3\|(k+s), u \text{ primitive in } \mathbb~~{F}~~_{2^n~~}^* </math>~~</td>~~		<td><math>n=2t</math> is even</td>
	~~<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions~~. ~~Cryptography~~ and ~~Communications~~. ~~2011 Mar 1;3(1~~)~~:43-53~~.</ref></td>		<td><ref name="kai_ref">>Tan Y, Qu L, Tan CH, Li C. New Families of Differentially 4-Uniform Permutations over <math>{\mathbb F} _ {2^{2k}} </math>. InInternational Conference on Sequences and Their Applications 2012 Jun 4 (pp. 25-39). Springer, Berlin, Heidelberg.</ref></td>
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	<td><math>~~(x+~~x^{~~2{^m}})^{2^k+~~1}+~~u'(ux+u^{2^~~{m}} x^{2^{m}}~~)^{(2^k~~+1)~~2^i~~}+u(x+x^{~~2^{m}~~})~~(ux+u~~^{2^{~~m}} x^{2^{m~~}})</math></td>		<td><math>x^{-1} + \mathrm {Tr}(x^{-3(2^{k}+1)}+ (x^{-1}+1)^{3(2^{k}+1)})</math></td>
	<td><math>n=~~2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1~~</math> and <math> ~~i \geqslant~~ 2~~</math> even, <math>u~~\~~text{ primitive in }~~ \~~mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }~~</math></td>		<td><math>n=2t</math> and <math>2\leq k \leq t-1</math></td>
	<td><ref~~>Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.<~~/~~ref~~></td>		<td><ref name="kai_ref" /></td>
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Fady at 11:29, 12 June 2019

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	<td><math>~~sx^{q+1}+~~x^{2^~~i+1~~}~~+x^{q(~~2^i+1~~)}+cx^{2^iq+1}+c^qx^{2^i+q~~}</math></td>		<td><math>x^{2^{2i}-2^i+1}</math></td>
	<td>~~<math>q=2^m, n=2m,</math>~~ <math>gcd(i,m)=~~1</math>, <math>c\in \mathbb{F}_{~~2~~^n}~~, ~~s \in \mathbb F_{2^~~n~~} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x~~</math> s.t~~. <math>x^{q+1}=1</math>~~</td>		<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
	<td><ref>~~Budaghyan L, Carlet C~~. ~~Classes~~ of ~~quadratic APN trinomials and hexanomials and related structures~~. ~~IEEE Transactions on~~ Information ~~Theory~~. ~~2008~~ May;54(5):~~2354~~-7.</ref></td>		<td><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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Fady at 11:20, 12 June 2019

2019-06-12T11:20:33Z

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	<td><math>x^{2^i+1}</math></td>		<td><math>x^{2^i+1}</math></td>
	<td><math>gcd(i,n) = 2, n = 2t and t is odd~~</math>~~</td>		<td><math>gcd(i,n) = 2, n = 2t</math> and t is odd</td>
	<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref></td>		<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref></td>
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Fady: Created page with "
2019-06-12T11:17:50Z

Created page with "<table> <tr> <th>Functions</th> <th>Conditions</th> <th>References</th> </tr> <tr> <td><math>x^{2^i+1}</math></td> <td><math>gcd(i,n) = 2, n = 2t and t is odd</math></td> <td..."

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<tr>
<th>Functions</th>
<th>Conditions</th>
<th>References</th>
</tr>

<tr>
<td><math>x^{2^i+1}</math></td>
<td><math>gcd(i,n) = 2, n = 2t and t is odd</math></td>
<td><ref>Gold R. Maximal recursive sequences with 3-valued recursive cross-correlation functions (Corresp.). IEEE transactions on Information Theory. 1968 Jan;14(1):154-6.</ref></td>
</tr>

<tr>
<td><math>sx^{q+1}+x^{2^i+1}+x^{q(2^i+1)}+cx^{2^iq+1}+c^qx^{2^i+q}</math></td>
<td><math>q=2^m, n=2m,</math> <math>gcd(i,m)=1</math>, <math>c\in \mathbb{F}_{2^n}, s \in \mathbb F_{2^n} \setminus \mathbb{F}_{q}, X^{2^i+1}+cX^{2^i}+c^{q}X+1 \text{ has no solution } x</math> s.t. <math>x^{q+1}=1</math></td>
<td><ref>Budaghyan L, Carlet C. Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Transactions on Information Theory. 2008 May;54(5):2354-7.</ref></td>
</td>
</tr>

<tr>
<td><math>x^3+a^{-1} \mathrm {Tr}_n (a^3x^9)</math></td>
<td><math>a\neq 0</math></td>
<td><ref>Budaghyan L, Carlet C, Leander G. Constructing new APN functions from known ones. Finite Fields and Their Applications. 2009 Apr 1;15(2):150-9.</ref></td>
</td>
</tr>

<tr>
<td><math>x^3+a^{-1} \mathrm {Tr}_n^3 (a^3x^9+a^6x^{18})</math></td>
<td><math>3|n </math>, <math>a\ne0</math></td>
<td><ref name="2_ref">Budaghyan L, Carlet C, Leander G. On a construction of quadratic APN functions. InInformation Theory Workshop, 2009. ITW 2009. IEEE 2009 Oct 11 (pp. 374-378). IEEE.</ref></td>
</tr>

<tr>
<td><math>x^3+a^{-1} \mathrm{Tr}_n^3(a^6x^{18}+a^{12}x^{36})</math></td>
<td> <math>3|n, a \ne 0</math></td>
<td><ref name="2_ref" /></td>
</tr>

<tr>
<td><math>ux^{2^s+1}+u^{2^k} x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+wu^{2^k+1}x^{2^{s}+2^{k+s}}</math></td>
<td><math>n=3k, \gcd(k,3)=\gcd(s,3k)=1, v, w\in\mathbb{F}_{2^k}, vw \ne 1, 3|(k+s), u \text{ primitive in } \mathbb{F}_{2^n}^* </math></td>
<td><ref>Bracken C, Byrne E, Markin N, Mcguire G. A few more quadratic APN functions. Cryptography and Communications. 2011 Mar 1;3(1):43-53.</ref></td>
</tr>

<tr>
<td><math>(x+x^{2{^m}})^{2^k+1}+u'(ux+u^{2^{m}} x^{2^{m}})^{(2^k+1)2^i}+u(x+x^{2^{m}})(ux+u^{2^{m}} x^{2^{m}})</math></td>
<td><math>n=2m, m\geqslant 2</math> even, <math>\gcd(k, m)=1</math> and <math> i \geqslant 2</math> even, <math>u\text{ primitive in } \mathbb{F}_{2^n}^*, u' \in \mathbb{F}_{2^m} \text{ not a cube }</math></td>
<td><ref>Zhou Y, Pott A. A new family of semifields with 2 parameters. Advances in Mathematics. 2013 Feb 15;234:43-60.</ref></td>
</tr>

<tr>
<td><math>a^2x^{2^{2m+1}+1}+b^2x^{2^{m+1}+1}+ax^{2^{2m}+2}+bx^{2^{m}+2}+(c^2+c)x^3</math></td>
<td><math>n=3m, m \ \text{odd}, L(x)=ax^{2^{2m}}+bx^{2^{m}}+cx</math> satisfies the conditions in Lemma 8 of [7]</td>
<td><ref>Villa I, Budaghyan L, Calderini M, Carlet C, & Coulter R. On Isotopic Construction of APN Functions. SETA 2018</ref></td>
</tr>
</table>

Functions Conditions References

$x^{2^i+1}$ $gcd(i,n) = 2, n = 2t and t is odd$