# Walsh spectra of all known APN functions over GF(2^8)

The tables below contain the Walsh spectra for all known instances of APN functions over GF(2^8). All of these 8180 functions have one of the following three Walsh spectra:

• ${\displaystyle \{-32^{2380},-16^{20400},0^{16320},16^{23120},32^{3060}\}}$ (same as the Gold functions)
• ${\displaystyle \{-64^{6},-32^{2240},-16^{20880},0^{15600},16^{23664},32^{2880},64^{10}\}}$ (type 1)
• ${\displaystyle \{-64^{12},-32^{2100},-16^{21360},0^{14880},16^{24208},32^{2700},64^{20}\}}$ (type 2)

There are 12 functions with a Walsh spectrum of type 2 (given in the table below), 487 functions with a Walsh spectrum of type 1, and 7681 functions with a Gold-like Walsh spectrum (not listed below due to space limitations). Magma code listing all functions with a Gold-like Walsh spectrum, a Walsh spectrum of type 1 and a Walsh spectrum of type 2 is available.

Functions with Walsh spectrum of type 2
${\displaystyle \alpha ^{130}\cdot x^{192}+\alpha ^{160}\cdot x^{160}+\alpha ^{117}\cdot x^{144}+\alpha ^{230}\cdot x^{136}+\alpha ^{228}\cdot x^{132}+\alpha ^{162}\cdot x^{130}+\alpha ^{25}\cdot x^{129}+\alpha ^{79}\cdot x^{96}+\alpha ^{204}\cdot x^{80}+\alpha ^{83}\cdot x^{72}+\alpha ^{159}\cdot x^{68}+\alpha ^{234}\cdot x^{66}+\alpha ^{36}\cdot x^{65}+\alpha ^{67}\cdot x^{48}+\alpha ^{151}\cdot x^{40}+\alpha ^{17}\cdot x^{36}+\alpha ^{81}\cdot x^{34}+\alpha ^{52}\cdot x^{33}+\alpha ^{9}\cdot x^{24}+\alpha ^{116}\cdot x^{20}+\alpha ^{102}\cdot x^{18}+\alpha ^{97}\cdot x^{17}+\alpha ^{74}\cdot x^{12}+\alpha ^{48}\cdot x^{10}+\alpha ^{144}\cdot x^{9}+\alpha ^{58}\cdot x^{6}+\alpha ^{146}\cdot x^{5}+\alpha ^{123}\cdot x^{3}}$
${\displaystyle \alpha ^{154}\cdot x^{192}+\alpha ^{36}\cdot x^{160}+\alpha ^{83}\cdot x^{144}+\alpha ^{160}\cdot x^{136}+\alpha ^{253}\cdot x^{132}+\alpha ^{215}\cdot x^{130}+\alpha ^{221}\cdot x^{129}+\alpha ^{76}\cdot x^{96}+\alpha ^{137}\cdot x^{80}+\alpha ^{206}\cdot x^{72}+\alpha ^{185}\cdot x^{68}+\alpha ^{165}\cdot x^{66}+\alpha ^{201}\cdot x^{65}+\alpha ^{226}\cdot x^{48}+\alpha ^{25}\cdot x^{40}+\alpha ^{65}\cdot x^{36}+\alpha ^{11}\cdot x^{33}+\alpha ^{170}\cdot x^{24}+\alpha ^{247}\cdot x^{20}+\alpha ^{155}\cdot x^{18}+\alpha \cdot x^{17}+\alpha ^{146}\cdot x^{12}+\alpha ^{204}\cdot x^{10}+\alpha ^{121}\cdot x^{9}+\alpha ^{202}\cdot x^{6}+\alpha ^{246}\cdot x^{5}+\alpha ^{170}\cdot x^{3}}$
${\displaystyle \alpha ^{183}\cdot x^{192}+\alpha ^{178}\cdot x^{160}+\alpha ^{103}\cdot x^{144}+\alpha ^{97}\cdot x^{136}+\alpha ^{37}\cdot x^{132}+\alpha ^{172}\cdot x^{130}+\alpha ^{102}\cdot x^{129}+\alpha ^{62}\cdot x^{96}+\alpha ^{145}\cdot x^{80}+\alpha ^{96}\cdot x^{72}+\alpha ^{132}\cdot x^{68}+\alpha ^{210}\cdot x^{66}+\alpha ^{69}\cdot x^{65}+\alpha ^{69}\cdot x^{48}+\alpha ^{11}\cdot x^{40}+x^{36}+\alpha ^{4}\cdot x^{34}+\alpha ^{76}\cdot x^{33}+\alpha ^{122}\cdot x^{24}+\alpha ^{6}\cdot x^{20}+\alpha ^{145}\cdot x^{18}+\alpha ^{155}\cdot x^{17}+\alpha ^{41}\cdot x^{12}+\alpha ^{40}\cdot x^{10}+\alpha ^{106}\cdot x^{9}+\alpha ^{144}\cdot x^{6}+\alpha ^{102}\cdot x^{5}+\alpha ^{246}\cdot x^{3}}$
${\displaystyle \alpha ^{22}\cdot x^{192}+\alpha ^{167}\cdot x^{160}+\alpha ^{178}\cdot x^{144}+\alpha ^{84}\cdot x^{136}+\alpha ^{219}\cdot x^{132}+\alpha ^{248}\cdot x^{130}+\alpha ^{130}\cdot x^{129}+\alpha ^{221}\cdot x^{96}+\alpha ^{84}\cdot x^{80}+\alpha ^{123}\cdot x^{72}+\alpha ^{140}\cdot x^{68}+\alpha ^{26}\cdot x^{66}+\alpha ^{108}\cdot x^{65}+\alpha ^{50}\cdot x^{48}+\alpha ^{15}\cdot x^{40}+\alpha ^{211}\cdot x^{36}+\alpha ^{116}\cdot x^{34}+\alpha ^{19}\cdot x^{33}+\alpha ^{228}\cdot x^{24}+\alpha ^{176}\cdot x^{20}+\alpha ^{42}\cdot x^{18}+\alpha ^{80}\cdot x^{17}+\alpha ^{180}\cdot x^{12}+\alpha ^{203}\cdot x^{10}+\alpha ^{104}\cdot x^{9}+\alpha ^{72}\cdot x^{6}+\alpha ^{151}\cdot x^{5}+\alpha ^{247}\cdot x^{3}}$
${\displaystyle \alpha ^{156}\cdot x^{192}+\alpha ^{25}\cdot x^{160}+\alpha ^{158}\cdot x^{144}+\alpha ^{20}\cdot x^{136}+\alpha ^{50}\cdot x^{132}+\alpha ^{140}\cdot x^{130}+\alpha ^{203}\cdot x^{129}+\alpha ^{184}\cdot x^{96}+\alpha ^{152}\cdot x^{80}+\alpha ^{228}\cdot x^{72}+\alpha ^{194}\cdot x^{68}+\alpha ^{203}\cdot x^{66}+\alpha ^{131}\cdot x^{65}+\alpha ^{25}\cdot x^{48}+\alpha ^{192}\cdot x^{40}+\alpha ^{191}\cdot x^{36}+\alpha ^{125}\cdot x^{34}+\alpha ^{136}\cdot x^{33}+\alpha ^{132}\cdot x^{24}+\alpha ^{85}\cdot x^{20}+\alpha ^{191}\cdot x^{18}+\alpha ^{120}\cdot x^{17}+\alpha ^{212}\cdot x^{12}+\alpha ^{244}\cdot x^{10}+\alpha ^{133}\cdot x^{9}+\alpha ^{78}\cdot x^{6}+\alpha ^{161}\cdot x^{5}+\alpha \cdot x^{3}}$
${\displaystyle \alpha ^{193}\cdot x^{192}+\alpha ^{33}\cdot x^{160}+\alpha ^{22}\cdot x^{144}+\alpha ^{204}\cdot x^{136}+\alpha ^{173}\cdot x^{132}+\alpha ^{50}\cdot x^{130}+\alpha ^{66}\cdot x^{129}+\alpha ^{42}\cdot x^{96}+\alpha ^{69}\cdot x^{80}+\alpha ^{175}\cdot x^{72}+\alpha ^{230}\cdot x^{68}+\alpha ^{253}\cdot x^{66}+\alpha ^{16}\cdot x^{65}+\alpha ^{52}\cdot x^{48}+\alpha ^{54}\cdot x^{40}+\alpha ^{9}\cdot x^{36}+\alpha ^{177}\cdot x^{34}+\alpha ^{99}\cdot x^{33}+\alpha ^{12}\cdot x^{24}+\alpha ^{37}\cdot x^{20}+\alpha ^{83}\cdot x^{18}+\alpha ^{230}\cdot x^{17}+\alpha ^{78}\cdot x^{12}+\alpha \cdot x^{10}+\alpha ^{64}\cdot x^{9}+\alpha ^{225}\cdot x^{6}+\alpha ^{68}\cdot x^{5}+\alpha ^{204}\cdot x^{3}}$
${\displaystyle \alpha ^{88}\cdot x^{192}+\alpha ^{8}\cdot x^{160}+\alpha ^{11}\cdot x^{144}+\alpha ^{121}\cdot x^{136}+\alpha ^{205}\cdot x^{132}+\alpha ^{165}\cdot x^{130}+\alpha ^{206}\cdot x^{129}+\alpha ^{164}\cdot x^{96}+\alpha ^{235}\cdot x^{80}+\alpha ^{94}\cdot x^{72}+\alpha ^{173}\cdot x^{68}+\alpha ^{142}\cdot x^{66}+\alpha ^{238}\cdot x^{65}+\alpha ^{102}\cdot x^{48}+\alpha ^{113}\cdot x^{40}+\alpha ^{183}\cdot x^{36}+\alpha ^{187}\cdot x^{34}+\alpha ^{157}\cdot x^{33}+\alpha ^{2}\cdot x^{24}+\alpha ^{23}\cdot x^{20}+\alpha ^{122}\cdot x^{18}+\alpha ^{21}\cdot x^{17}+\alpha ^{154}\cdot x^{12}+\alpha ^{78}\cdot x^{10}+\alpha ^{117}\cdot x^{9}+\alpha ^{177}\cdot x^{6}+\alpha ^{111}\cdot x^{5}+\alpha ^{60}\cdot x^{3}}$
${\displaystyle \alpha ^{212}\cdot x^{192}+\alpha ^{198}\cdot x^{160}+\alpha ^{175}\cdot x^{144}+\alpha ^{80}\cdot x^{136}+\alpha ^{196}\cdot x^{132}+\alpha ^{167}\cdot x^{130}+\alpha ^{2}\cdot x^{129}+\alpha ^{65}\cdot x^{96}+\alpha ^{243}\cdot x^{80}+\alpha ^{91}\cdot x^{72}+\alpha ^{171}\cdot x^{68}+\alpha ^{211}\cdot x^{66}+\alpha ^{182}\cdot x^{65}+\alpha ^{247}\cdot x^{48}+\alpha ^{86}\cdot x^{40}+\alpha ^{89}\cdot x^{36}+\alpha ^{87}\cdot x^{34}+\alpha ^{83}\cdot x^{33}+\alpha ^{138}\cdot x^{24}+\alpha ^{45}\cdot x^{20}+\alpha ^{149}\cdot x^{18}+\alpha ^{100}\cdot x^{17}+\alpha ^{188}\cdot x^{12}+\alpha ^{17}\cdot x^{10}+\alpha ^{243}\cdot x^{9}+\alpha ^{237}\cdot x^{6}+\alpha ^{112}\cdot x^{5}+\alpha ^{137}\cdot x^{3}}$
${\displaystyle \alpha ^{117}\cdot x^{192}+\alpha ^{61}\cdot x^{160}+\alpha ^{230}\cdot x^{144}+\alpha ^{105}\cdot x^{136}+\alpha ^{191}\cdot x^{132}+\alpha ^{113}\cdot x^{130}+\alpha ^{245}\cdot x^{129}+\alpha ^{139}\cdot x^{96}+\alpha ^{166}\cdot x^{80}+\alpha ^{210}\cdot x^{72}+\alpha ^{221}\cdot x^{68}+\alpha ^{138}\cdot x^{66}+\alpha ^{146}\cdot x^{65}+\alpha ^{120}\cdot x^{48}+\alpha ^{124}\cdot x^{40}+\alpha ^{252}\cdot x^{36}+\alpha ^{182}\cdot x^{34}+\alpha ^{5}\cdot x^{33}+\alpha ^{8}\cdot x^{24}+\alpha ^{136}\cdot x^{20}+\alpha ^{235}\cdot x^{18}+\alpha ^{61}\cdot x^{17}+\alpha ^{45}\cdot x^{12}+\alpha ^{149}\cdot x^{10}+\alpha ^{158}\cdot x^{9}+\alpha ^{13}\cdot x^{6}+\alpha ^{169}\cdot x^{5}+\alpha ^{121}\cdot x^{3}}$
${\displaystyle \alpha ^{34}\cdot x^{192}+\alpha ^{57}\cdot x^{160}+\alpha ^{187}\cdot x^{144}+\alpha ^{36}\cdot x^{136}+\alpha ^{137}\cdot x^{132}+\alpha ^{63}\cdot x^{130}+\alpha ^{98}\cdot x^{129}+\alpha ^{236}\cdot x^{96}+\alpha ^{161}\cdot x^{80}+\alpha ^{66}\cdot x^{72}+\alpha ^{191}\cdot x^{68}+\alpha ^{117}\cdot x^{66}+\alpha ^{241}\cdot x^{65}+\alpha ^{7}\cdot x^{48}+\alpha ^{9}\cdot x^{40}+\alpha ^{153}\cdot x^{36}+\alpha ^{118}\cdot x^{34}+\alpha ^{154}\cdot x^{33}+\alpha ^{194}\cdot x^{24}+\alpha ^{157}\cdot x^{20}+\alpha ^{14}\cdot x^{18}+\alpha ^{116}\cdot x^{17}+\alpha ^{119}\cdot x^{12}+\alpha ^{113}\cdot x^{10}+\alpha ^{13}\cdot x^{9}+\alpha ^{138}\cdot x^{6}+\alpha ^{143}\cdot x^{5}+\alpha ^{35}\cdot x^{3}}$
${\displaystyle \alpha ^{140}\cdot x^{192}+\alpha ^{233}\cdot x^{160}+\alpha ^{150}\cdot x^{144}+\alpha ^{146}\cdot x^{136}+\alpha ^{99}\cdot x^{132}+\alpha ^{249}\cdot x^{130}+\alpha ^{211}\cdot x^{129}+\alpha ^{66}\cdot x^{96}+\alpha ^{37}\cdot x^{80}+\alpha ^{35}\cdot x^{72}+\alpha ^{199}\cdot x^{68}+\alpha ^{170}\cdot x^{66}+\alpha ^{2}\cdot x^{65}+\alpha ^{217}\cdot x^{48}+\alpha ^{2}\cdot x^{40}+\alpha ^{192}\cdot x^{36}+\alpha ^{32}\cdot x^{34}+\alpha ^{229}\cdot x^{33}+\alpha ^{241}\cdot x^{24}+\alpha ^{200}\cdot x^{20}+\alpha ^{63}\cdot x^{18}+\alpha ^{17}\cdot x^{17}+\alpha ^{251}\cdot x^{12}+\alpha ^{44}\cdot x^{10}+\alpha ^{106}\cdot x^{9}+\alpha ^{25}\cdot x^{6}+\alpha ^{174}\cdot x^{5}+\alpha ^{127}\cdot x^{3}}$
${\displaystyle \alpha ^{237}\cdot x^{192}+\alpha ^{133}\cdot x^{160}+\alpha ^{204}\cdot x^{144}+\alpha ^{169}\cdot x^{136}+\alpha ^{30}\cdot x^{132}+\alpha ^{127}\cdot x^{130}+\alpha ^{41}\cdot x^{129}+\alpha ^{12}\cdot x^{96}+\alpha ^{198}\cdot x^{80}+\alpha ^{151}\cdot x^{72}+\alpha ^{252}\cdot x^{68}+\alpha ^{29}\cdot x^{66}+\alpha ^{144}\cdot x^{65}+\alpha ^{120}\cdot x^{48}+\alpha ^{72}\cdot x^{40}+\alpha ^{123}\cdot x^{36}+\alpha ^{170}\cdot x^{34}+\alpha ^{159}\cdot x^{33}+\alpha ^{77}\cdot x^{24}+\alpha ^{227}\cdot x^{20}+\alpha ^{161}\cdot x^{18}+\alpha ^{231}\cdot x^{17}+\alpha ^{159}\cdot x^{12}+\alpha ^{253}\cdot x^{10}+\alpha ^{56}\cdot x^{9}+\alpha ^{35}\cdot x^{6}+\alpha ^{251}\cdot x^{5}+\alpha ^{99}\cdot x^{3}}$