Difference between revisions of "Walsh spectra of all known APN functions over GF(2^8)"

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(Created page with "The tables below contain the Walsh spectra for all quadratic APN functions of dimension 8 as given in the appendix of [https://eprint.iacr.org/2013/007.pdf "A Matrix Approach...")
 
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* <math>\{ -64^6, -32^{2240}, -16^{20880}, 0^{15600}, 16^{23664}, 32^{2880}, 64^{10} \}</math> (type 1)
 
* <math>\{ -64^6, -32^{2240}, -16^{20880}, 0^{15600}, 16^{23664}, 32^{2880}, 64^{10} \}</math> (type 1)
 
* <math>\{ -64^{12}, -32^{2100}, -16^{21360}, 0^{14880}, 16^{24208}, 32^{2700}, 64^{20} \}</math> (type 2)
 
* <math>\{ -64^{12}, -32^{2100}, -16^{21360}, 0^{14880}, 16^{24208}, 32^{2700}, 64^{20} \}</math> (type 2)
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<table>
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<tr>
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<th>Functions with Walsh spectrum of type 2</th>
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</tr>
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<tr>
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<td><math>\alpha^{130}\cdot x^{192} + \alpha^{160}\cdot x^{160} + \alpha^{117}\cdot x^{144} + \alpha^{230}\cdot x^{136} + \alpha^{228}\cdot x^{132} + \alpha^{162}\cdot x^{130} + \alpha^{25}\cdot x^{129} + \alpha^{79}\cdot x^{96} + \alpha^{204}\cdot x^{80} + \alpha^{83}\cdot x^{72} + \alpha^{159}\cdot x^{68} + \alpha^{234}\cdot x^{66} + \alpha^{36}\cdot x^{65} + \alpha^{67}\cdot x^{48} + \alpha^{151}\cdot x^{40} + \alpha^{17}\cdot x^{36} + \alpha^{81}\cdot x^{34} + \alpha^{52}\cdot x^{33} + \alpha^{9}\cdot x^{24} + \alpha^{116}\cdot x^{20} + \alpha^{102}\cdot x^{18} + \alpha^{97}\cdot x^{17} + \alpha^{74}\cdot x^{12} + \alpha^{48}\cdot x^{10} + \alpha^{144}\cdot x^{9} + \alpha^{58}\cdot x^{6} + \alpha^{146}\cdot x^{5} + \alpha^{123}\cdot x^{3}</math></td>
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<td><math>\alpha^{154}\cdot x^{192} + \alpha^{36}\cdot x^{160} + \alpha^{83}\cdot x^{144} + \alpha^{160}\cdot x^{136} + \alpha^{253}\cdot x^{132} + \alpha^{215}\cdot x^{130} + \alpha^{221}\cdot x^{129} + \alpha^{76}\cdot x^{96} + \alpha^{137}\cdot x^{80} + \alpha^{206}\cdot x^{72} + \alpha^{185}\cdot x^{68} + \alpha^{165}\cdot x^{66} + \alpha^{201}\cdot x^{65} + \alpha^{226}\cdot x^{48} + \alpha^{25}\cdot x^{40} + \alpha^{65}\cdot x^{36} + \alpha^{11}\cdot x^{33} + \alpha^{170}\cdot x^{24} + \alpha^{247}\cdot x^{20} + \alpha^{155}\cdot x^{18} + \alpha\cdot x^{17} + \alpha^{146}\cdot x^{12} + \alpha^{204}\cdot x^{10} + \alpha^{121}\cdot x^{9} + \alpha^{202}\cdot x^{6} + \alpha^{246}\cdot x^{5} + \alpha^{170}\cdot x^{3}</math></td>
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<td><math>\alpha^{183}\cdot x^{192} + \alpha^{178}\cdot x^{160} + \alpha^{103}\cdot x^{144} + \alpha^{97}\cdot x^{136} + \alpha^{37}\cdot x^{132} + \alpha^{172}\cdot x^{130} + \alpha^{102}\cdot x^{129} + \alpha^{62}\cdot x^{96} + \alpha^{145}\cdot x^{80} + \alpha^{96}\cdot x^{72} + \alpha^{132}\cdot x^{68} + \alpha^{210}\cdot x^{66} + \alpha^{69}\cdot x^{65} + \alpha^{69}\cdot x^{48} + \alpha^{11}\cdot x^{40} + x^{36} + \alpha^{4}\cdot x^{34} + \alpha^{76}\cdot x^{33} + \alpha^{122}\cdot x^{24} + \alpha^{6}\cdot x^{20} + \alpha^{145}\cdot x^{18} + \alpha^{155}\cdot x^{17} + \alpha^{41}\cdot x^{12} + \alpha^{40}\cdot x^{10} + \alpha^{106}\cdot x^{9} + \alpha^{144}\cdot x^{6} + \alpha^{102}\cdot x^{5} + \alpha^{246}\cdot x^{3}</math></td>
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<td><math>\alpha^{22}\cdot x^{192} + \alpha^{167}\cdot x^{160} + \alpha^{178}\cdot x^{144} + \alpha^{84}\cdot x^{136} + \alpha^{219}\cdot x^{132} + \alpha^{248}\cdot x^{130} + \alpha^{130}\cdot x^{129} + \alpha^{221}\cdot x^{96} + \alpha^{84}\cdot x^{80} + \alpha^{123}\cdot x^{72} + \alpha^{140}\cdot x^{68} + \alpha^{26}\cdot x^{66} + \alpha^{108}\cdot x^{65} + \alpha^{50}\cdot x^{48} + \alpha^{15}\cdot x^{40} + \alpha^{211}\cdot x^{36} + \alpha^{116}\cdot x^{34} + \alpha^{19}\cdot x^{33} + \alpha^{228}\cdot x^{24} + \alpha^{176}\cdot x^{20} + \alpha^{42}\cdot x^{18} + \alpha^{80}\cdot x^{17} + \alpha^{180}\cdot x^{12} + \alpha^{203}\cdot x^{10} + \alpha^{104}\cdot x^{9} + \alpha^{72}\cdot x^{6} + \alpha^{151}\cdot x^{5} + \alpha^{247}\cdot x^{3}</math></td>
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<td><math>\alpha^{156}\cdot x^{192} + \alpha^{25}\cdot x^{160} + \alpha^{158}\cdot x^{144} + \alpha^{20}\cdot x^{136} + \alpha^{50}\cdot x^{132} + \alpha^{140}\cdot x^{130} + \alpha^{203}\cdot x^{129} + \alpha^{184}\cdot x^{96} + \alpha^{152}\cdot x^{80} + \alpha^{228}\cdot x^{72} + \alpha^{194}\cdot x^{68} + \alpha^{203}\cdot x^{66} + \alpha^{131}\cdot x^{65} + \alpha^{25}\cdot x^{48} + \alpha^{192}\cdot x^{40} + \alpha^{191}\cdot x^{36} + \alpha^{125}\cdot x^{34} + \alpha^{136}\cdot x^{33} + \alpha^{132}\cdot x^{24} + \alpha^{85}\cdot x^{20} + \alpha^{191}\cdot x^{18} + \alpha^{120}\cdot x^{17} + \alpha^{212}\cdot x^{12} + \alpha^{244}\cdot x^{10} + \alpha^{133}\cdot x^{9} + \alpha^{78}\cdot x^{6} + \alpha^{161}\cdot x^{5} + \alpha\cdot x^{3}</math></td>
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<td><math>\alpha^{193}\cdot x^{192} + \alpha^{33}\cdot x^{160} + \alpha^{22}\cdot x^{144} + \alpha^{204}\cdot x^{136} + \alpha^{173}\cdot x^{132} + \alpha^{50}\cdot x^{130} + \alpha^{66}\cdot x^{129} + \alpha^{42}\cdot x^{96} + \alpha^{69}\cdot x^{80} + \alpha^{175}\cdot x^{72} + \alpha^{230}\cdot x^{68} + \alpha^{253}\cdot x^{66} + \alpha^{16}\cdot x^{65} + \alpha^{52}\cdot x^{48} + \alpha^{54}\cdot x^{40} + \alpha^{9}\cdot x^{36} + \alpha^{177}\cdot x^{34} + \alpha^{99}\cdot x^{33} + \alpha^{12}\cdot x^{24} + \alpha^{37}\cdot x^{20} + \alpha^{83}\cdot x^{18} + \alpha^{230}\cdot x^{17} + \alpha^{78}\cdot x^{12} + \alpha\cdot x^{10} + \alpha^{64}\cdot x^{9} + \alpha^{225}\cdot x^{6} + \alpha^{68}\cdot x^{5} + \alpha^{204}\cdot x^{3}</math></td>
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<td><math>\alpha^{88}\cdot x^{192} + \alpha^{8}\cdot x^{160} + \alpha^{11}\cdot x^{144} + \alpha^{121}\cdot x^{136} + \alpha^{205}\cdot x^{132} + \alpha^{165}\cdot x^{130} + \alpha^{206}\cdot x^{129} + \alpha^{164}\cdot x^{96} + \alpha^{235}\cdot x^{80} + \alpha^{94}\cdot x^{72} + \alpha^{173}\cdot x^{68} + \alpha^{142}\cdot x^{66} + \alpha^{238}\cdot x^{65} + \alpha^{102}\cdot x^{48} + \alpha^{113}\cdot x^{40} + \alpha^{183}\cdot x^{36} + \alpha^{187}\cdot x^{34} + \alpha^{157}\cdot x^{33} + \alpha^{2}\cdot x^{24} + \alpha^{23}\cdot x^{20} + \alpha^{122}\cdot x^{18} + \alpha^{21}\cdot x^{17} + \alpha^{154}\cdot x^{12} + \alpha^{78}\cdot x^{10} + \alpha^{117}\cdot x^{9} + \alpha^{177}\cdot x^{6} + \alpha^{111}\cdot x^{5} + \alpha^{60}\cdot x^{3}</math></td>
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<td><math>\alpha^{212}\cdot x^{192} + \alpha^{198}\cdot x^{160} + \alpha^{175}\cdot x^{144} + \alpha^{80}\cdot x^{136} + \alpha^{196}\cdot x^{132} + \alpha^{167}\cdot x^{130} + \alpha^{2}\cdot x^{129} + \alpha^{65}\cdot x^{96} + \alpha^{243}\cdot x^{80} + \alpha^{91}\cdot x^{72} + \alpha^{171}\cdot x^{68} + \alpha^{211}\cdot x^{66} + \alpha^{182}\cdot x^{65} + \alpha^{247}\cdot x^{48} + \alpha^{86}\cdot x^{40} + \alpha^{89}\cdot x^{36} + \alpha^{87}\cdot x^{34} + \alpha^{83}\cdot x^{33} + \alpha^{138}\cdot x^{24} + \alpha^{45}\cdot x^{20} + \alpha^{149}\cdot x^{18} + \alpha^{100}\cdot x^{17} + \alpha^{188}\cdot x^{12} + \alpha^{17}\cdot x^{10} + \alpha^{243}\cdot x^{9} + \alpha^{237}\cdot x^{6} + \alpha^{112}\cdot x^{5} + \alpha^{137}\cdot x^{3}</math></td>
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</tr>
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<td><math>\alpha^{117}\cdot x^{192} + \alpha^{61}\cdot x^{160} + \alpha^{230}\cdot x^{144} + \alpha^{105}\cdot x^{136} + \alpha^{191}\cdot x^{132} + \alpha^{113}\cdot x^{130} + \alpha^{245}\cdot x^{129} + \alpha^{139}\cdot x^{96} + \alpha^{166}\cdot x^{80} + \alpha^{210}\cdot x^{72} + \alpha^{221}\cdot x^{68} + \alpha^{138}\cdot x^{66} + \alpha^{146}\cdot x^{65} + \alpha^{120}\cdot x^{48} + \alpha^{124}\cdot x^{40} + \alpha^{252}\cdot x^{36} + \alpha^{182}\cdot x^{34} + \alpha^{5}\cdot x^{33} + \alpha^{8}\cdot x^{24} + \alpha^{136}\cdot x^{20} + \alpha^{235}\cdot x^{18} + \alpha^{61}\cdot x^{17} + \alpha^{45}\cdot x^{12} + \alpha^{149}\cdot x^{10} + \alpha^{158}\cdot x^{9} + \alpha^{13}\cdot x^{6} + \alpha^{169}\cdot x^{5} + \alpha^{121}\cdot x^{3}</math></td>
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<td><math>\alpha^{34}\cdot x^{192} + \alpha^{57}\cdot x^{160} + \alpha^{187}\cdot x^{144} + \alpha^{36}\cdot x^{136} + \alpha^{137}\cdot x^{132} + \alpha^{63}\cdot x^{130} + \alpha^{98}\cdot x^{129} + \alpha^{236}\cdot x^{96} + \alpha^{161}\cdot x^{80} + \alpha^{66}\cdot x^{72} + \alpha^{191}\cdot x^{68} + \alpha^{117}\cdot x^{66} + \alpha^{241}\cdot x^{65} + \alpha^{7}\cdot x^{48} + \alpha^{9}\cdot x^{40} + \alpha^{153}\cdot x^{36} + \alpha^{118}\cdot x^{34} + \alpha^{154}\cdot x^{33} + \alpha^{194}\cdot x^{24} + \alpha^{157}\cdot x^{20} + \alpha^{14}\cdot x^{18} + \alpha^{116}\cdot x^{17} + \alpha^{119}\cdot x^{12} + \alpha^{113}\cdot x^{10} + \alpha^{13}\cdot x^{9} + \alpha^{138}\cdot x^{6} + \alpha^{143}\cdot x^{5} + \alpha^{35}\cdot x^{3}</math></td>
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<td><math>\alpha^{140}\cdot x^{192} + \alpha^{233}\cdot x^{160} + \alpha^{150}\cdot x^{144} + \alpha^{146}\cdot x^{136} + \alpha^{99}\cdot x^{132} + \alpha^{249}\cdot x^{130} + \alpha^{211}\cdot x^{129} + \alpha^{66}\cdot x^{96} + \alpha^{37}\cdot x^{80} + \alpha^{35}\cdot x^{72} + \alpha^{199}\cdot x^{68} + \alpha^{170}\cdot x^{66} + \alpha^{2}\cdot x^{65} + \alpha^{217}\cdot x^{48} + \alpha^{2}\cdot x^{40} + \alpha^{192}\cdot x^{36} + \alpha^{32}\cdot x^{34} + \alpha^{229}\cdot x^{33} + \alpha^{241}\cdot x^{24} + \alpha^{200}\cdot x^{20} + \alpha^{63}\cdot x^{18} + \alpha^{17}\cdot x^{17} + \alpha^{251}\cdot x^{12} + \alpha^{44}\cdot x^{10} + \alpha^{106}\cdot x^{9} + \alpha^{25}\cdot x^{6} + \alpha^{174}\cdot x^{5} + \alpha^{127}\cdot x^{3}</math></td>
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<td><math>\alpha^{237}\cdot x^{192} + \alpha^{133}\cdot x^{160} + \alpha^{204}\cdot x^{144} + \alpha^{169}\cdot x^{136} + \alpha^{30}\cdot x^{132} + \alpha^{127}\cdot x^{130} + \alpha^{41}\cdot x^{129} + \alpha^{12}\cdot x^{96} + \alpha^{198}\cdot x^{80} + \alpha^{151}\cdot x^{72} + \alpha^{252}\cdot x^{68} + \alpha^{29}\cdot x^{66} + \alpha^{144}\cdot x^{65} + \alpha^{120}\cdot x^{48} + \alpha^{72}\cdot x^{40} + \alpha^{123}\cdot x^{36} + \alpha^{170}\cdot x^{34} + \alpha^{159}\cdot x^{33} + \alpha^{77}\cdot x^{24} + \alpha^{227}\cdot x^{20} + \alpha^{161}\cdot x^{18} + \alpha^{231}\cdot x^{17} + \alpha^{159}\cdot x^{12} + \alpha^{253}\cdot x^{10} + \alpha^{56}\cdot x^{9} + \alpha^{35}\cdot x^{6} + \alpha^{251}\cdot x^{5} + \alpha^{99}\cdot x^{3}</math></td>
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</table>

Revision as of 17:45, 19 December 2018

The tables below contain the Walsh spectra for all quadratic APN functions of dimension 8 as given in the appendix of "A Matrix Approach for Constructing Quadratic APN Functions". All 8180 functions listed in the appendix have one of the following three Walsh spectra:

  • (same as the Gold functions)
  • (type 1)
  • (type 2)


Functions with Walsh spectrum of type 2