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

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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:
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The tables below contain the Walsh spectra for all [[known instances of APN functions over GF(2^8)]]. All of these 8192 functions have one of the following three Walsh spectra:
* <math>\{ -32^{2380}, -16^{20400}, 0^{16320}, 16^{23120}, 32^{3060} \}</math> (same as the Gold functions)
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* <span class="htmlMath">{ -32<sup>2380</sup>, -16<sup>20400</sup>, 0<sup>16320</sup>, 16<sup>23120</sup>, 32<sup>3060</sup> }</span> (same as the Gold functions)
* <math>\{ -64^6, -32^{2240}, -16^{20880}, 0^{15600}, 16^{23664}, 32^{2880}, 64^{10} \}</math> (type 1)
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* <span class="htmlMath">{ -64<sup>6</sup>, -32<sup>2240</sup>, -16<sup>20880</sup>, 0<sup>15600</sup>, 16<sup>23664</sup>, 32<sup>2880</sup>, 64<sup>10</sup> }</span> (type 1)
* <math>\{ -64^{12}, -32^{2100}, -16^{21360}, 0^{14880}, 16^{24208}, 32^{2700}, 64^{20} \}</math> (type 2)
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* <span class="htmlMath">{ -64<sup>12</sup>, -32<sup>2100</sup>, -16<sup>21360</sup>, 0<sup>14880</sup>, 16<sup>24208</sup>, 32<sup>2700</sup>, 64<sup>20</sup> }</span> (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 [[:File:Gold spectra.txt|Gold-like Walsh spectrum]], a [[:File:T1 spectra.txt|Walsh spectrum of type 1]] and a [[:File:T2 spectra.txt|Walsh spectrum of type 2]] is available.
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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 7693 functions with a Gold-like Walsh spectrum (not listed below due to space limitations). Magma code listing all functions with a [[:File:Gold spectra.txt|Gold-like Walsh spectrum]], a [[:File:T1 spectra.txt|Walsh spectrum of type 1]] and a [[:File:T2 spectra.txt|Walsh spectrum of type 2]] is available.
  
  
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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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<span class="htmlMath">g<sup>130</sup>x<sup>192</sup> + g<sup>160</sup>x<sup>160</sup> + g<sup>117</sup>x<sup>144</sup> + g<sup>230</sup>x<sup>136</sup> + g<sup>228</sup>x<sup>132</sup> + g<sup>162</sup>x<sup>130</sup> + g<sup>25</sup>x<sup>129</sup> + g<sup>79</sup>x<sup>96</sup> + g<sup>204</sup>x<sup>80</sup> + g<sup>83</sup>x<sup>72</sup> + g<sup>159</sup>x<sup>68</sup> + g<sup>234</sup>x<sup>66</sup> + g<sup>36</sup>x<sup>65</sup> + g<sup>67</sup>x<sup>48</sup> + g<sup>151</sup>x<sup>40</sup> + g<sup>17</sup>x<sup>36</sup> + g<sup>81</sup>x<sup>34</sup> + g<sup>52</sup>x<sup>33</sup> + g<sup>9</sup>x<sup>24</sup> + g<sup>116</sup>x<sup>20</sup> + g<sup>102</sup>x<sup>18</sup> + g<sup>97</sup>x<sup>17</sup> + g<sup>74</sup>x<sup>12</sup> + g<sup>48</sup>x<sup>10</sup> + g<sup>144</sup>x<sup>9</sup> + g<sup>58</sup>x<sup>6</sup> + g<sup>146</sup>x<sup>5</sup> + g<sup>123</sup>x<sup>3</sup></span></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><span class="htmlMath">g<sup>154</sup>x<sup>192</sup> + g<sup>36</sup>x<sup>160</sup> + g<sup>83</sup>x<sup>144</sup> + g<sup>160</sup>x<sup>136</sup> + g<sup>253</sup>x<sup>132</sup> + g<sup>215</sup>x<sup>130</sup> + g<sup>221</sup>x<sup>129</sup> + g<sup>76</sup>x<sup>96</sup> + g<sup>137</sup>x<sup>80</sup> + g<sup>206</sup>x<sup>72</sup> + g<sup>185</sup>x<sup>68</sup> + g<sup>165</sup>x<sup>66</sup> + g<sup>201</sup>x<sup>65</sup> + g<sup>226</sup>x<sup>48</sup> + g<sup>25</sup>x<sup>40</sup> + g<sup>65</sup>x<sup>36</sup> + g<sup>11</sup>x<sup>33</sup> + g<sup>170</sup>x<sup>24</sup> + g<sup>247</sup>x<sup>20</sup> + g<sup>155</sup>x<sup>18</sup> + gx<sup>17</sup> + g<sup>146</sup>x<sup>12</sup> + g<sup>204</sup>x<sup>10</sup> + g<sup>121</sup>x<sup>9</sup> + g<sup>202</sup>x<sup>6</sup> + g<sup>246</sup>x<sup>5</sup> + g<sup>170</sup>x<sup>3</sup></span></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><span class="htmlMath">g<sup>183</sup>x<sup>192</sup> + g<sup>178</sup>x<sup>160</sup> + g<sup>103</sup>x<sup>144</sup> + g<sup>97</sup>x<sup>136</sup> + g<sup>37</sup>x<sup>132</sup> + g<sup>172</sup>x<sup>130</sup> + g<sup>102</sup>x<sup>129</sup> + g<sup>62</sup>x<sup>96</sup> + g<sup>145</sup>x<sup>80</sup> + g<sup>96</sup>x<sup>72</sup> + g<sup>132</sup>x<sup>68</sup> + g<sup>210</sup>x<sup>66</sup> + g<sup>69</sup>x<sup>65</sup> + g<sup>69</sup>x<sup>48</sup> + g<sup>11</sup>x<sup>40</sup> +x<sup>36</sup> + g<sup>4</sup>x<sup>34</sup> + g<sup>76</sup>x<sup>33</sup> + g<sup>122</sup>x<sup>24</sup> + g<sup>6</sup>x<sup>20</sup> + g<sup>145</sup>x<sup>18</sup> + g<sup>155</sup>x<sup>17</sup> + g<sup>41</sup>x<sup>12</sup> + g<sup>40</sup>x<sup>10</sup> + g<sup>106</sup>x<sup>9</sup> + g<sup>144</sup>x<sup>6</sup> + g<sup>102</sup>x<sup>5</sup> + g<sup>246</sup>x<sup>3</sup></span></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><span class="htmlMath">g<sup>22</sup>x<sup>192</sup> + g<sup>167</sup>x<sup>160</sup> + g<sup>178</sup>x<sup>144</sup> + g<sup>84</sup>x<sup>136</sup> + g<sup>219</sup>x<sup>132</sup> + g<sup>248</sup>x<sup>130</sup> + g<sup>130</sup>x<sup>129</sup> + g<sup>221</sup>x<sup>96</sup> + g<sup>84</sup>x<sup>80</sup> + g<sup>123</sup>x<sup>72</sup> + g<sup>140</sup>x<sup>68</sup> + g<sup>26</sup>x<sup>66</sup> + g<sup>108</sup>x<sup>65</sup> + g<sup>50</sup>x<sup>48</sup> + g<sup>15</sup>x<sup>40</sup> + g<sup>211</sup>x<sup>36</sup> + g<sup>116</sup>x<sup>34</sup> + g<sup>19</sup>x<sup>33</sup> + g<sup>228</sup>x<sup>24</sup> + g<sup>176</sup>x<sup>20</sup> + g<sup>42</sup>x<sup>18</sup> + g<sup>80</sup>x<sup>17</sup> + g<sup>180</sup>x<sup>12</sup> + g<sup>203</sup>x<sup>10</sup> + g<sup>104</sup>x<sup>9</sup> + g<sup>72</sup>x<sup>6</sup> + g<sup>151</sup>x<sup>5</sup> + g<sup>247</sup>x<sup>3</sup></span></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><span class="htmlMath">g<sup>156</sup>x<sup>192</sup> + g<sup>25</sup>x<sup>160</sup> + g<sup>158</sup>x<sup>144</sup> + g<sup>20</sup>x<sup>136</sup> + g<sup>50</sup>x<sup>132</sup> + g<sup>140</sup>x<sup>130</sup> + g<sup>203</sup>x<sup>129</sup> + g<sup>184</sup>x<sup>96</sup> + g<sup>152</sup>x<sup>80</sup> + g<sup>228</sup>x<sup>72</sup> + g<sup>194</sup>x<sup>68</sup> + g<sup>203</sup>x<sup>66</sup> + g<sup>131</sup>x<sup>65</sup> + g<sup>25</sup>x<sup>48</sup> + g<sup>192</sup>x<sup>40</sup> + g<sup>191</sup>x<sup>36</sup> + g<sup>125</sup>x<sup>34</sup> + g<sup>136</sup>x<sup>33</sup> + g<sup>132</sup>x<sup>24</sup> + g<sup>85</sup>x<sup>20</sup> + g<sup>191</sup>x<sup>18</sup> + g<sup>120</sup>x<sup>17</sup> + g<sup>212</sup>x<sup>12</sup> + g<sup>244</sup>x<sup>10</sup> + g<sup>133</sup>x<sup>9</sup> + g<sup>78</sup>x<sup>6</sup> + g<sup>161</sup>x<sup>5</sup> + gx<sup>3</sup></span></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><span class="htmlMath">g<sup>193</sup>x<sup>192</sup> + g<sup>33</sup>x<sup>160</sup> + g<sup>22</sup>x<sup>144</sup> + g<sup>204</sup>x<sup>136</sup> + g<sup>173</sup>x<sup>132</sup> + g<sup>50</sup>x<sup>130</sup> + g<sup>66</sup>x<sup>129</sup> + g<sup>42</sup>x<sup>96</sup> + g<sup>69</sup>x<sup>80</sup> + g<sup>175</sup>x<sup>72</sup> + g<sup>230</sup>x<sup>68</sup> + g<sup>253</sup>x<sup>66</sup> + g<sup>16</sup>x<sup>65</sup> + g<sup>52</sup>x<sup>48</sup> + g<sup>54</sup>x<sup>40</sup> + g<sup>9</sup>x<sup>36</sup> + g<sup>177</sup>x<sup>34</sup> + g<sup>99</sup>x<sup>33</sup> + g<sup>12</sup>x<sup>24</sup> + g<sup>37</sup>x<sup>20</sup> + g<sup>83</sup>x<sup>18</sup> + g<sup>230</sup>x<sup>17</sup> + g<sup>78</sup>x<sup>12</sup> + gx<sup>10</sup> + g<sup>64</sup>x<sup>9</sup> + g<sup>225</sup>x<sup>6</sup> + g<sup>68</sup>x<sup>5</sup> + g<sup>204</sup>x<sup>3</sup></span></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><span class="htmlMath">g<sup>88</sup>x<sup>192</sup> + g<sup>8</sup>x<sup>160</sup> + g<sup>11</sup>x<sup>144</sup> + g<sup>121</sup>x<sup>136</sup> + g<sup>205</sup>x<sup>132</sup> + g<sup>165</sup>x<sup>130</sup> + g<sup>206</sup>x<sup>129</sup> + g<sup>164</sup>x<sup>96</sup> + g<sup>235</sup>x<sup>80</sup> + g<sup>94</sup>x<sup>72</sup> + g<sup>173</sup>x<sup>68</sup> + g<sup>142</sup>x<sup>66</sup> + g<sup>238</sup>x<sup>65</sup> + g<sup>102</sup>x<sup>48</sup> + g<sup>113</sup>x<sup>40</sup> + g<sup>183</sup>x<sup>36</sup> + g<sup>187</sup>x<sup>34</sup> + g<sup>157</sup>x<sup>33</sup> + g<sup>2</sup>x<sup>24</sup> + g<sup>23</sup>x<sup>20</sup> + g<sup>122</sup>x<sup>18</sup> + g<sup>21</sup>x<sup>17</sup> + g<sup>154</sup>x<sup>12</sup> + g<sup>78</sup>x<sup>10</sup> + g<sup>117</sup>x<sup>9</sup> + g<sup>177</sup>x<sup>6</sup> + g<sup>111</sup>x<sup>5</sup> + g<sup>60</sup>x<sup>3</sup></span></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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<td><span class="htmlMath">g<sup>212</sup>x<sup>192</sup> + g<sup>198</sup>x<sup>160</sup> + g<sup>175</sup>x<sup>144</sup> + g<sup>80</sup>x<sup>136</sup> + g<sup>196</sup>x<sup>132</sup> + g<sup>167</sup>x<sup>130</sup> + g<sup>2</sup>x<sup>129</sup> + g<sup>65</sup>x<sup>96</sup> + g<sup>243</sup>x<sup>80</sup> + g<sup>91</sup>x<sup>72</sup> + g<sup>171</sup>x<sup>68</sup> + g<sup>211</sup>x<sup>66</sup> + g<sup>182</sup>x<sup>65</sup> + g<sup>247</sup>x<sup>48</sup> + g<sup>86</sup>x<sup>40</sup> + g<sup>89</sup>x<sup>36</sup> + g<sup>87</sup>x<sup>34</sup> + g<sup>83</sup>x<sup>33</sup> + g<sup>138</sup>x<sup>24</sup> + g<sup>45</sup>x<sup>20</sup> + g<sup>149</sup>x<sup>18</sup> + g<sup>100</sup>x<sup>17</sup> + g<sup>188</sup>x<sup>12</sup> + g<sup>17</sup>x<sup>10</sup> + g<sup>243</sup>x<sup>9</sup> + g<sup>237</sup>x<sup>6</sup> + g<sup>112</sup>x<sup>5</sup> + g<sup>137</sup>x<sup>3</sup></span></td>
 
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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>
+
<td><span class="htmlMath">g<sup>117</sup>x<sup>192</sup> + g<sup>61</sup>x<sup>160</sup> + g<sup>230</sup>x<sup>144</sup> + g<sup>105</sup>x<sup>136</sup> + g<sup>191</sup>x<sup>132</sup> + g<sup>113</sup>x<sup>130</sup> + g<sup>245</sup>x<sup>129</sup> + g<sup>139</sup>x<sup>96</sup> + g<sup>166</sup>x<sup>80</sup> + g<sup>210</sup>x<sup>72</sup> + g<sup>221</sup>x<sup>68</sup> + g<sup>138</sup>x<sup>66</sup> + g<sup>146</sup>x<sup>65</sup> + g<sup>120</sup>x<sup>48</sup> + g<sup>124</sup>x<sup>40</sup> + g<sup>252</sup>x<sup>36</sup> + g<sup>182</sup>x<sup>34</sup> + g<sup>5</sup>x<sup>33</sup> + g<sup>8</sup>x<sup>24</sup> + g<sup>136</sup>x<sup>20</sup> + g<sup>235</sup>x<sup>18</sup> + g<sup>61</sup>x<sup>17</sup> + g<sup>45</sup>x<sup>12</sup> + g<sup>149</sup>x<sup>10</sup> + g<sup>158</sup>x<sup>9</sup> + g<sup>13</sup>x<sup>6</sup> + g<sup>169</sup>x<sup>5</sup> + g<sup>121</sup>x<sup>3</sup></span></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
<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>
+
<td><span class="htmlMath">g<sup>34</sup>x<sup>192</sup> + g<sup>57</sup>x<sup>160</sup> + g<sup>187</sup>x<sup>144</sup> + g<sup>36</sup>x<sup>136</sup> + g<sup>137</sup>x<sup>132</sup> + g<sup>63</sup>x<sup>130</sup> + g<sup>98</sup>x<sup>129</sup> + g<sup>236</sup>x<sup>96</sup> + g<sup>161</sup>x<sup>80</sup> + g<sup>66</sup>x<sup>72</sup> + g<sup>191</sup>x<sup>68</sup> + g<sup>117</sup>x<sup>66</sup> + g<sup>241</sup>x<sup>65</sup> + g<sup>7</sup>x<sup>48</sup> + g<sup>9</sup>x<sup>40</sup> + g<sup>153</sup>x<sup>36</sup> + g<sup>118</sup>x<sup>34</sup> + g<sup>154</sup>x<sup>33</sup> + g<sup>194</sup>x<sup>24</sup> + g<sup>157</sup>x<sup>20</sup> + g<sup>14</sup>x<sup>18</sup> + g<sup>116</sup>x<sup>17</sup> + g<sup>119</sup>x<sup>12</sup> + g<sup>113</sup>x<sup>10</sup> + g<sup>13</sup>x<sup>9</sup> + g<sup>138</sup>x<sup>6</sup> + g<sup>143</sup>x<sup>5</sup> + g<sup>35</sup>x<sup>3</sup></span></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
<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>
+
<td><span class="htmlMath">g<sup>140</sup>x<sup>192</sup> + g<sup>233</sup>x<sup>160</sup> + g<sup>150</sup>x<sup>144</sup> + g<sup>146</sup>x<sup>136</sup> + g<sup>99</sup>x<sup>132</sup> + g<sup>249</sup>x<sup>130</sup> + g<sup>211</sup>x<sup>129</sup> + g<sup>66</sup>x<sup>96</sup> + g<sup>37</sup>x<sup>80</sup> + g<sup>35</sup>x<sup>72</sup> + g<sup>199</sup>x<sup>68</sup> + g<sup>170</sup>x<sup>66</sup> + g<sup>2</sup>x<sup>65</sup> + g<sup>217</sup>x<sup>48</sup> + g<sup>2</sup>x<sup>40</sup> + g<sup>192</sup>x<sup>36</sup> + g<sup>32</sup>x<sup>34</sup> + g<sup>229</sup>x<sup>33</sup> + g<sup>241</sup>x<sup>24</sup> + g<sup>200</sup>x<sup>20</sup> + g<sup>63</sup>x<sup>18</sup> + g<sup>17</sup>x<sup>17</sup> + g<sup>251</sup>x<sup>12</sup> + g<sup>44</sup>x<sup>10</sup> + g<sup>106</sup>x<sup>9</sup> + g<sup>25</sup>x<sup>6</sup> + g<sup>174</sup>x<sup>5</sup> + g<sup>127</sup>x<sup>3</sup></span></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
<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>
+
<td><span class="htmlMath">g<sup>237</sup>x<sup>192</sup> + g<sup>133</sup>x<sup>160</sup> + g<sup>204</sup>x<sup>144</sup> + g<sup>169</sup>x<sup>136</sup> + g<sup>30</sup>x<sup>132</sup> + g<sup>127</sup>x<sup>130</sup> + g<sup>41</sup>x<sup>129</sup> + g<sup>12</sup>x<sup>96</sup> + g<sup>198</sup>x<sup>80</sup> + g<sup>151</sup>x<sup>72</sup> + g<sup>252</sup>x<sup>68</sup> + g<sup>29</sup>x<sup>66</sup> + g<sup>144</sup>x<sup>65</sup> + g<sup>120</sup>x<sup>48</sup> + g<sup>72</sup>x<sup>40</sup> + g<sup>123</sup>x<sup>36</sup> + g<sup>170</sup>x<sup>34</sup> + g<sup>159</sup>x<sup>33</sup> + g<sup>77</sup>x<sup>24</sup> + g<sup>227</sup>x<sup>20</sup> + g<sup>161</sup>x<sup>18</sup> + g<sup>231</sup>x<sup>17</sup> + g<sup>159</sup>x<sup>12</sup> + g<sup>253</sup>x<sup>10</sup> + g<sup>56</sup>x<sup>9</sup> + g<sup>35</sup>x<sup>6</sup> + g<sup>251</sup>x<sup>5</sup> + g<sup>99</sup>x<sup>3</sup></span></td>
 
</tr>
 
</tr>
 
</table>
 
</table>

Latest revision as of 15:35, 14 January 2021

The tables below contain the Walsh spectra for all known instances of APN functions over GF(2^8). All of these 8192 functions have one of the following three Walsh spectra:

  • { -322380, -1620400, 016320, 1623120, 323060 } (same as the Gold functions)
  • { -646, -322240, -1620880, 015600, 1623664, 322880, 6410 } (type 1)
  • { -6412, -322100, -1621360, 014880, 1624208, 322700, 6420 } (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 7693 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
g130x192 + g160x160 + g117x144 + g230x136 + g228x132 + g162x130 + g25x129 + g79x96 + g204x80 + g83x72 + g159x68 + g234x66 + g36x65 + g67x48 + g151x40 + g17x36 + g81x34 + g52x33 + g9x24 + g116x20 + g102x18 + g97x17 + g74x12 + g48x10 + g144x9 + g58x6 + g146x5 + g123x3
g154x192 + g36x160 + g83x144 + g160x136 + g253x132 + g215x130 + g221x129 + g76x96 + g137x80 + g206x72 + g185x68 + g165x66 + g201x65 + g226x48 + g25x40 + g65x36 + g11x33 + g170x24 + g247x20 + g155x18 + gx17 + g146x12 + g204x10 + g121x9 + g202x6 + g246x5 + g170x3
g183x192 + g178x160 + g103x144 + g97x136 + g37x132 + g172x130 + g102x129 + g62x96 + g145x80 + g96x72 + g132x68 + g210x66 + g69x65 + g69x48 + g11x40 +x36 + g4x34 + g76x33 + g122x24 + g6x20 + g145x18 + g155x17 + g41x12 + g40x10 + g106x9 + g144x6 + g102x5 + g246x3
g22x192 + g167x160 + g178x144 + g84x136 + g219x132 + g248x130 + g130x129 + g221x96 + g84x80 + g123x72 + g140x68 + g26x66 + g108x65 + g50x48 + g15x40 + g211x36 + g116x34 + g19x33 + g228x24 + g176x20 + g42x18 + g80x17 + g180x12 + g203x10 + g104x9 + g72x6 + g151x5 + g247x3
g156x192 + g25x160 + g158x144 + g20x136 + g50x132 + g140x130 + g203x129 + g184x96 + g152x80 + g228x72 + g194x68 + g203x66 + g131x65 + g25x48 + g192x40 + g191x36 + g125x34 + g136x33 + g132x24 + g85x20 + g191x18 + g120x17 + g212x12 + g244x10 + g133x9 + g78x6 + g161x5 + gx3
g193x192 + g33x160 + g22x144 + g204x136 + g173x132 + g50x130 + g66x129 + g42x96 + g69x80 + g175x72 + g230x68 + g253x66 + g16x65 + g52x48 + g54x40 + g9x36 + g177x34 + g99x33 + g12x24 + g37x20 + g83x18 + g230x17 + g78x12 + gx10 + g64x9 + g225x6 + g68x5 + g204x3
g88x192 + g8x160 + g11x144 + g121x136 + g205x132 + g165x130 + g206x129 + g164x96 + g235x80 + g94x72 + g173x68 + g142x66 + g238x65 + g102x48 + g113x40 + g183x36 + g187x34 + g157x33 + g2x24 + g23x20 + g122x18 + g21x17 + g154x12 + g78x10 + g117x9 + g177x6 + g111x5 + g60x3
g212x192 + g198x160 + g175x144 + g80x136 + g196x132 + g167x130 + g2x129 + g65x96 + g243x80 + g91x72 + g171x68 + g211x66 + g182x65 + g247x48 + g86x40 + g89x36 + g87x34 + g83x33 + g138x24 + g45x20 + g149x18 + g100x17 + g188x12 + g17x10 + g243x9 + g237x6 + g112x5 + g137x3
g117x192 + g61x160 + g230x144 + g105x136 + g191x132 + g113x130 + g245x129 + g139x96 + g166x80 + g210x72 + g221x68 + g138x66 + g146x65 + g120x48 + g124x40 + g252x36 + g182x34 + g5x33 + g8x24 + g136x20 + g235x18 + g61x17 + g45x12 + g149x10 + g158x9 + g13x6 + g169x5 + g121x3
g34x192 + g57x160 + g187x144 + g36x136 + g137x132 + g63x130 + g98x129 + g236x96 + g161x80 + g66x72 + g191x68 + g117x66 + g241x65 + g7x48 + g9x40 + g153x36 + g118x34 + g154x33 + g194x24 + g157x20 + g14x18 + g116x17 + g119x12 + g113x10 + g13x9 + g138x6 + g143x5 + g35x3
g140x192 + g233x160 + g150x144 + g146x136 + g99x132 + g249x130 + g211x129 + g66x96 + g37x80 + g35x72 + g199x68 + g170x66 + g2x65 + g217x48 + g2x40 + g192x36 + g32x34 + g229x33 + g241x24 + g200x20 + g63x18 + g17x17 + g251x12 + g44x10 + g106x9 + g25x6 + g174x5 + g127x3
g237x192 + g133x160 + g204x144 + g169x136 + g30x132 + g127x130 + g41x129 + g12x96 + g198x80 + g151x72 + g252x68 + g29x66 + g144x65 + g120x48 + g72x40 + g123x36 + g170x34 + g159x33 + g77x24 + g227x20 + g161x18 + g231x17 + g159x12 + g253x10 + g56x9 + g35x6 + g251x5 + g99x3