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→Finding a Single Byte: Finished
== Finding a Single Byte ==
Okay, we know that our power trace will look a lot different for one of our choices of signatures. Let's figure out which one. We'll start by finding the average over all of our 1000 traces:
<pre>
# Find the average over all of the traces
mean = np.average(traces, axis=0)
</pre>
Then, we'll split our traces into 256 different groups (one for each plaintext). Since we know the IV, we can now use it to recover the actual plaintext that the bootloader checks:
<pre>
# Split the traces into groups
groupedTraces = [[] for _ in range(256)]
for i in range(numTraces):
group = dr[i][0] ^ 0xC1
groupedTraces[group].append(traces[i])
</pre>
Next, we can find the mean for each group and see how much they differ from the overall mean:
<pre>
# Find the mean for each group
means = np.zeros([256, traceLen])
for i in range(256):
if len(groupedTraces[i]) > 0:
means[i] = np.average(groupedTraces[i], axis=0)
plt.plot(means[0] - mean)
plt.plot(means[1] - mean)
plt.plot(means[2] - mean)
plt.plot(means[3] - mean)
plt.grid()
plt.show()
</pre>
The plot that comes out of this should look a bit like:
[[File:Tutorial-A5-Bonus-Signature.PNG]]
Wow - looks like we found it! However, let's clean this up with some statistics. We can use the correlation coefficient to see which bytes are the furthest away from the average:
<pre>
corr = []
for i in range(256):
corr.append(np.corrcoef(mean[1500:1700], means[i][1500:1700])[0, 1])
print np.sort(corr)
print np.argsort(corr)
</pre>
This should print something that looks like:
<pre>
[ 0.67663819 0.9834704 0.98855246 0.98942869 0.98994226 0.99019698
0.99082752 0.99159262 0.99166859 0.99169598 0.99216366 0.99229359
0.99239152 0.99240231 0.99246389 0.99254908 0.99258285 0.9926239
0.99280577 0.99302107 0.99339631 0.99394492 0.99396965 0.99403114
0.99408231 0.99410649 0.99424916 0.99460312 0.99464971 0.99468856
0.99483947 0.99512576 0.99553707 0.99570373 0.99572752 0.99577311
0.99582226 0.99587666 0.99590395 0.99623462 0.99630861 0.99639056
0.99644546 0.99646228 0.99653183 0.99661089 0.9966309 0.99665822
0.9966832 0.99670105 0.99673815 0.99679397 0.99690167 0.99692316
0.9969269 0.99694459 0.99703105 0.99704228 0.99705158 0.99708642
0.99709179 0.9971024 0.99710707 0.99711091 0.99711536 0.99715928
0.99720665 0.99721363 0.99721902 0.99722437 0.99722547 0.99723478
0.99724198 0.997244 0.99724712 0.99728416 0.99728622 0.99729196
0.99734564 0.99737952 0.99739401 0.99742793 0.99745246 0.99747648
0.99750044 0.9975651 0.99760837 0.99762965 0.99763106 0.99763222
0.99765327 0.9976662 0.9976953 0.99769761 0.99771007 0.99773553
0.99775314 0.99777414 0.99782335 0.99785114 0.99786062 0.99787688
0.99788584 0.99788938 0.9978924 0.99793722 0.99797874 0.99798273
0.9980249 0.99807047 0.99807947 0.99810194 0.99813208 0.9982722
0.99838807 0.99843216 0.99856034 0.99856295 0.99863064 0.9987529
0.99878124 0.99882028 0.99884917 0.99890103 0.99890116 0.99890879
0.99891135 0.99891317 0.99893291 0.99893508 0.99894488 0.99894848
0.99897892 0.99898304 0.9989834 0.99898804 0.99901833 0.99905207
0.99905642 0.99905798 0.99908281 0.99910538 0.99911272 0.99911782
0.99912193 0.99912223 0.9991229 0.99914415 0.99914732 0.99916885
0.99917188 0.99917941 0.99918178 0.99919009 0.99921141 0.99923463
0.99924823 0.99924986 0.99925438 0.99925524 0.99926407 0.99927205
0.99927364 0.99928305 0.99928533 0.99929447 0.99929925 0.99930205
0.99930243 0.99930623 0.99931579 0.99932861 0.99933414 0.99933806
0.99933992 0.99934213 0.99935681 0.99935721 0.9993594 0.9993601
0.99936267 0.99936373 0.99936482 0.99937458 0.99937665 0.99937706
0.99938049 0.99938241 0.99938251 0.999391 0.99940622 0.9994087
0.99940929 0.9994159 0.99941886 0.99942033 0.99942274 0.99942601
0.9994279 0.99943674 0.99943796 0.99944123 0.99944152 0.99944193
0.99944859 0.9994499 0.99945661 0.9994776 0.99948316 0.99949018
0.9994928 0.99949457 0.99949475 0.99949542 0.99949547 0.99949835
0.99950941 0.99951938 0.99951941 0.99953141 0.9995379 0.99954004
0.99954337 0.99954548 0.99955606 0.9995565 0.99956179 0.99956494
0.99956494 0.99956716 0.99957014 0.99957477 0.99957663 0.99958413
0.99958574 0.99958651 0.99958795 0.99958879 0.99959042 0.99959141
0.99959237 0.99959677 0.99961313 0.99962923 0.99963177 0.9996504
0.99968832 0.99969333 0.99969583 0.99969834 0.99970998 0.99972495
0.99972646 nan nan nan]
[ 0 32 128 255 160 223 8 16 48 96 40 1 95 215 2 33 34 64
4 36 127 207 239 254 253 247 222 251 159 191 221 219 80 129 136 176
168 192 144 56 224 162 130 119 87 72 132 24 126 9 17 111 123 18
112 68 63 125 79 3 66 93 94 49 42 161 237 206 31 35 104 20
98 245 37 238 10 65 52 50 246 231 243 44 41 183 5 6 214 97
190 12 250 220 91 175 199 252 205 249 189 151 235 143 218 157 158 213
203 38 100 211 187 217 155 55 200 226 11 107 138 120 152 23 103 137
81 145 25 30 118 109 110 60 7 184 202 146 117 21 225 177 131 208
77 148 78 193 71 85 140 196 133 47 185 115 15 86 233 169 61 172
194 232 122 186 62 92 102 75 124 212 116 29 150 180 156 57 230 121
90 182 240 167 76 170 165 88 43 229 166 46 147 27 188 163 149 19
198 51 210 53 73 83 142 135 59 114 22 197 241 45 236 227 89 174
82 67 13 244 14 181 228 69 195 58 39 26 242 173 113 74 179 141
106 99 234 105 216 28 139 153 209 201 204 248 54 108 84 171 101 70
154 164 134 178]
</pre>
This output tells us two things:
* The first list says that almost every trace looks very similar to the overall mean (98% correlated or higher). However, there's one trace that is totally different, with 68% correlation. This is probably our correct guess.
* The second list gives the signature guess that matches each of the above correlations. The first number in the list is 0x00, which is the correct signature!
Note that three numbers in this output show a correlation of <code>nan</code> because none of the captured traces had any data on them. However, this doesn't matter to us - we found our byte.
To finish this attack, you could force the capture software to send more specific text. To find the next byte of the signature, you'd want to fix byte 0 at 0x00 and make byte 1 random. Then, the plaintext should be XORed with the known IV and encrypted with the known AES-256 key. This is left as an exercise for the reader.
= Appendix A: IV Attack Script =