Part 4, Topic 2: CPA on Firmware Implementation of AES¶
SUMMARY: By now, you'll have used a DPA attack to break AES. While this method has its place in side channel attacks, it often requires a large number of traces to break AES and can suffer from additional issues like ghost peaks.
We've also learned in the previous lab that there is a very linear relationship between the hamming weight of the SBox output and the power consumption at that point. Instead of checking average power consumption over many traces to see if a guessed subkey is correct, we can instead check if our guessed subkey also has this linear relationship with the device's power consumption across a set of traces. Like with DPA, we'll need to repeat this measurement at each point in time along the power trace.
To get an objective measurement of how linear this relationship is, we'll be developing some code to calculate the Pearson correlation coefficient.
LEARNING OUTCOMES:
- Developing an algorithm based on a mathematical description
- Verify that correlation can be used to break a single byte of AES
- Extend the single byte attack to the rest of the key
Prerequisites¶
This notebook will build upon previous ones. Make sure you've completed the following tutorials and their prerequisites:
- ☑ Part 3 notebooks (you should be comfortable with running an attack on AES)
- ☑ Power and Hamming Weight Relationship (we'll be using information from this tutorial)
AES Trace Capture¶
Our first step will be to send some plaintext to the target device and observe its power consumption during the encryption. The capture loop will be the same as in the DPA attack. This time, however, we'll only need 50 traces to recover the key, a major improvement over the last attack!
Depending what you are using, you can complete this either by:
- Capturing new traces from a physical device.
- Reading pre-recorded data from a file.
You get to choose your adventure - see the two notebooks with the same name of this, but called (SIMULATED) or (HARDWARE) to continue. Inside those notebooks you should get some code to copy into the following section, which will define the capture function.
Be sure you get the "✔️ OK to continue!" print once you run the next cell, otherwise things will fail later on!
SCOPETYPE = 'OPENADC'
PLATFORM = 'CW308_SAM4S'
CRYPTO_TARGET = 'TINYAES128C'
VERSION = 'HARDWARE'
allowable_exceptions = None
SS_VER = 'SS_VER_2_1'
LONG_TEST = 'No'
if VERSION == 'HARDWARE':
#!/usr/bin/env python
# coding: utf-8
# # Part 4, Topic 2: CPA on Firmware Implementation of AES (HARDWARE)
# ---
# **THIS IS NOT THE COMPLETE TUTORIAL - see file with `(MAIN)` in the name.**
#
# ---
# First you'll need to select which hardware setup you have. You'll need to select a `SCOPETYPE`, a `PLATFORM`, and a `CRYPTO_TARGET`. `SCOPETYPE` can either be `'OPENADC'` for the CWLite/CW1200 or `'CWNANO'` for the CWNano. `PLATFORM` is the target device, with `'CWLITEARM'`/`'CW308_STM32F3'` being the best supported option, followed by `'CWLITEXMEGA'`/`'CW308_XMEGA'`, then by `'CWNANO'`. `CRYPTO_TARGET` selects the crypto implementation, with `'TINYAES128C'` working on all platforms. An alternative for `'CWLITEXMEGA'` targets is `'AVRCRYPTOLIB'`. For example:
#
# ```python
# SCOPETYPE = 'OPENADC'
# PLATFORM = 'CWLITEARM'
# CRYPTO_TARGET='TINYAES128C'
# SS_VER='SS_VER_2_1'
# ```
# In[ ]:
# The following code will build the firmware for the target.
# In[ ]:
#!/usr/bin/env python
# coding: utf-8
# In[ ]:
import chipwhisperer as cw
try:
if not scope.connectStatus:
scope.con()
except NameError:
scope = cw.scope(hw_location=(5, 8))
try:
if SS_VER == "SS_VER_2_1":
target_type = cw.targets.SimpleSerial2
elif SS_VER == "SS_VER_2_0":
raise OSError("SS_VER_2_0 is deprecated. Use SS_VER_2_1")
else:
target_type = cw.targets.SimpleSerial
except:
SS_VER="SS_VER_1_1"
target_type = cw.targets.SimpleSerial
try:
target = cw.target(scope, target_type)
except:
print("INFO: Caught exception on reconnecting to target - attempting to reconnect to scope first.")
print("INFO: This is a work-around when USB has died without Python knowing. Ignore errors above this line.")
scope = cw.scope(hw_location=(5, 8))
target = cw.target(scope, target_type)
print("INFO: Found ChipWhisperer😍")
# In[ ]:
if "STM" in PLATFORM or PLATFORM == "CWLITEARM" or PLATFORM == "CWNANO":
prog = cw.programmers.STM32FProgrammer
elif PLATFORM == "CW303" or PLATFORM == "CWLITEXMEGA":
prog = cw.programmers.XMEGAProgrammer
elif "neorv32" in PLATFORM.lower():
prog = cw.programmers.NEORV32Programmer
elif PLATFORM == "CW308_SAM4S" or PLATFORM == "CWHUSKY":
prog = cw.programmers.SAM4SProgrammer
else:
prog = None
# In[ ]:
import time
time.sleep(0.05)
scope.default_setup()
def reset_target(scope):
if PLATFORM == "CW303" or PLATFORM == "CWLITEXMEGA":
scope.io.pdic = 'low'
time.sleep(0.1)
scope.io.pdic = 'high_z' #XMEGA doesn't like pdic driven high
time.sleep(0.1) #xmega needs more startup time
elif "neorv32" in PLATFORM.lower():
raise IOError("Default iCE40 neorv32 build does not have external reset - reprogram device to reset")
elif PLATFORM == "CW308_SAM4S" or PLATFORM == "CWHUSKY":
scope.io.nrst = 'low'
time.sleep(0.25)
scope.io.nrst = 'high_z'
time.sleep(0.25)
else:
scope.io.nrst = 'low'
time.sleep(0.05)
scope.io.nrst = 'high_z'
time.sleep(0.05)
# In[ ]:
try:
get_ipython().run_cell_magic('bash', '-s "$PLATFORM" "$CRYPTO_TARGET" "$SS_VER"', 'cd ../../../firmware/mcu/simpleserial-aes\nmake PLATFORM=$1 CRYPTO_TARGET=$2 SS_VER=$3 -j\n &> /tmp/tmp.txt')
except:
x=open("/tmp/tmp.txt").read(); print(x); raise OSError(x)
# In[ ]:
cw.program_target(scope, prog, "../../../firmware/mcu/simpleserial-aes/simpleserial-aes-{}.hex".format(PLATFORM))
# We only need 50 traces this time to break AES!
# In[ ]:
splot = cw.StreamPlot()
splot.plot()
# In[ ]:
from tqdm.notebook import trange
import numpy as np
import time
ktp = cw.ktp.Basic()
trace_array = []
textin_array = []
key, text = ktp.next()
target.set_key(key)
N = 50 #increase nano reliability
if PLATFORM=="CWNANO":
N = 200
for i in trange(N, desc='Capturing traces'):
scope.arm()
target.simpleserial_write('p', text)
ret = scope.capture()
if ret:
print("Target timed out!")
continue
response = target.simpleserial_read('r', 16)
trace_array.append(scope.get_last_trace())
textin_array.append(text)
key, text = ktp.next()
splot.update(scope.get_last_trace())
trace_array = np.array(trace_array)
# We don't need the hardware anymore, so we'll disconnect:
# In[ ]:
scope.dis()
target.dis()
elif VERSION == 'SIMULATED':
#!/usr/bin/env python
# coding: utf-8
# # Part 4, Topic 2: CPA on Firmware Implementation of AES (SIMULATED)
# ---
# **THIS IS NOT THE COMPLETE TUTORIAL - see file with `(MAIN)` in the name.**
#
# ---
# Instead of performing a capture - just copy this data into the referenced code block. It is a copy of the previously recorded data.
# In[ ]:
from cwtraces import sca101_lab_data
import chipwhisperer as cw
data = sca101_lab_data["lab4_2"]()
trace_array = data["trace_array"]
textin_array = data["textin_array"]
key = data["key"]
INFO: Found ChipWhisperer😍
scope.gain.mode changed from low to high scope.gain.gain changed from 0 to 22 scope.gain.db changed from 15.0 to 25.091743119266056 scope.adc.samples changed from 131124 to 5000 scope.clock.clkgen\_freq changed from 0 to 7370129.87012987 scope.clock.adc\_freq changed from 0 to 29480519.48051948 scope.clock.extclk\_monitor\_enabled changed from True to False scope.clock.extclk\_tolerance changed from 1144409.1796875 to 13096723.705530167 scope.io.tio1 changed from serial\_tx to serial\_rx scope.io.tio2 changed from serial\_rx to serial\_tx scope.io.hs2 changed from None to clkgen scope.glitch.phase\_shift\_steps changed from 0 to 4592 scope.trace.capture.trigger\_source changed from trace trigger, rule #0 to firmware trigger Building for platform CW308\_SAM4S with CRYPTO\_TARGET=TINYAES128C
SS\_VER set to SS\_VER\_2\_1
SS\_VER set to SS\_VER\_2\_1
Blank crypto options, building for AES128
.
arm-none-eabi-gcc (15:9-2019-q4-0ubuntu1) 9.2.1 20191025 (release) [ARM/arm-9-branch revision 277599]
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
Welcome to another exciting ChipWhisperer target build!!
Size after:
+--------------------------------------------------------
+ Built for platform Microchip SAM4S with:
text data bss dec hex filename
3744 272 4520 8536 2158 simpleserial-aes-CW308\_SAM4S.elf
+ CRYPTO\_TARGET = TINYAES128C
+ CRYPTO\_OPTIONS = AES128C
+--------------------------------------------------------
Again, let's quickly plot a trace to make sure everything looks as expected:
# ###################
# START SOLUTION
# ###################
cw.plot(trace_array[0]) * cw.plot(trace_array[1])
# ###################
# END SOLUTION
# ###################
AES Model and Hamming Weight¶
Like with the previous tutorial, we'll need to be able to easily grab what the sbox output will be for a given plaintext and key, as well as get the hamming weight of numbers between 0 and 255:
# ###################
# Add your code here
# ###################
#raise NotImplementedError("Add your code here, and delete this.")
# ###################
# START SOLUTION
# ###################
sbox = [
# 0 1 2 3 4 5 6 7 8 9 a b c d e f
0x63,0x7c,0x77,0x7b,0xf2,0x6b,0x6f,0xc5,0x30,0x01,0x67,0x2b,0xfe,0xd7,0xab,0x76, # 0
0xca,0x82,0xc9,0x7d,0xfa,0x59,0x47,0xf0,0xad,0xd4,0xa2,0xaf,0x9c,0xa4,0x72,0xc0, # 1
0xb7,0xfd,0x93,0x26,0x36,0x3f,0xf7,0xcc,0x34,0xa5,0xe5,0xf1,0x71,0xd8,0x31,0x15, # 2
0x04,0xc7,0x23,0xc3,0x18,0x96,0x05,0x9a,0x07,0x12,0x80,0xe2,0xeb,0x27,0xb2,0x75, # 3
0x09,0x83,0x2c,0x1a,0x1b,0x6e,0x5a,0xa0,0x52,0x3b,0xd6,0xb3,0x29,0xe3,0x2f,0x84, # 4
0x53,0xd1,0x00,0xed,0x20,0xfc,0xb1,0x5b,0x6a,0xcb,0xbe,0x39,0x4a,0x4c,0x58,0xcf, # 5
0xd0,0xef,0xaa,0xfb,0x43,0x4d,0x33,0x85,0x45,0xf9,0x02,0x7f,0x50,0x3c,0x9f,0xa8, # 6
0x51,0xa3,0x40,0x8f,0x92,0x9d,0x38,0xf5,0xbc,0xb6,0xda,0x21,0x10,0xff,0xf3,0xd2, # 7
0xcd,0x0c,0x13,0xec,0x5f,0x97,0x44,0x17,0xc4,0xa7,0x7e,0x3d,0x64,0x5d,0x19,0x73, # 8
0x60,0x81,0x4f,0xdc,0x22,0x2a,0x90,0x88,0x46,0xee,0xb8,0x14,0xde,0x5e,0x0b,0xdb, # 9
0xe0,0x32,0x3a,0x0a,0x49,0x06,0x24,0x5c,0xc2,0xd3,0xac,0x62,0x91,0x95,0xe4,0x79, # a
0xe7,0xc8,0x37,0x6d,0x8d,0xd5,0x4e,0xa9,0x6c,0x56,0xf4,0xea,0x65,0x7a,0xae,0x08, # b
0xba,0x78,0x25,0x2e,0x1c,0xa6,0xb4,0xc6,0xe8,0xdd,0x74,0x1f,0x4b,0xbd,0x8b,0x8a, # c
0x70,0x3e,0xb5,0x66,0x48,0x03,0xf6,0x0e,0x61,0x35,0x57,0xb9,0x86,0xc1,0x1d,0x9e, # d
0xe1,0xf8,0x98,0x11,0x69,0xd9,0x8e,0x94,0x9b,0x1e,0x87,0xe9,0xce,0x55,0x28,0xdf, # e
0x8c,0xa1,0x89,0x0d,0xbf,0xe6,0x42,0x68,0x41,0x99,0x2d,0x0f,0xb0,0x54,0xbb,0x16 # f
]
def aes_internal(inputdata, key):
return sbox[inputdata ^ key]
HW = [bin(n).count("1") for n in range(0, 256)]
# ###################
# END SOLUTION
# ###################
Verify that your model is correct:
assert HW[aes_internal(0xA1, 0x79)] == 3
assert HW[aes_internal(0x22, 0xB1)] == 5
print("✔️ OK to continue!")
✔️ OK to continue!
Developing our Correlation Algorithm¶
As we discussed earlier, we'll be testing how good our guess is using a measurement called the Pearson correlation coefficient, which measures the linear correlation between two datasets.
The actual algorithm is as follows for datasets $X$ and $Y$ of length $N$, with means of $\bar{X}$ and $\bar{Y}$, respectively:
$$r = \frac{cov(X, Y)}{\sigma_X \sigma_Y}$$
$cov(X, Y)$ is the covariance of X and Y and can be calculated as follows:
$$cov(X, Y) = \sum_{n=1}^{N}[(Y_n - \bar{Y})(X_n - \bar{X})]$$
$\sigma_X$ and $\sigma_Y$ are the standard deviation of the two datasets. This value can be calculated with the following equation:
$$\sigma_X = \sqrt{\sum_{n=1}^{N}(X_n - \bar{X})^2}$$
As you can see, the calulation is actually broken down pretty nicely into some smaller chunks that we can implement with some simple functions. While we could use a library to calculate all this stuff for us, being able to implement a mathematical algorithm in code is a useful skill to develop.
To start, build the following functions:
mean(X)to calculate the mean of a datasetstd_dev(X, X_bar)to calculate the standard deviation of a dataset. We'll need to reuse the mean for the covariance, so it makes more sense to calculate it once and pass it in to each functioncov(X, X_bar, Y, Y_bar)to calculate the covariance of two datasets. Again, we can just pass in the means we calculate for std_dev here.
HINT: You can use np.sum(X, axis=0) to replace all of the $\sum$ from earlier. The argument axis=0 will sum across columns, allowing us to use a single mean, std_dev, and cov call for the entire power trace
# ###################
# Add your code here
# ###################
#raise NotImplementedError("Add your code here, and delete this.")
# ###################
# START SOLUTION
# ###################
def mean(X):
return np.sum(X, axis=0)/len(X)
def std_dev(X, X_bar):
return np.sqrt(np.sum((X-X_bar)**2, axis=0))
def cov(X, X_bar, Y, Y_bar):
return np.sum((X-X_bar)*(Y-Y_bar), axis=0)
# ###################
# END SOLUTION
# ###################
Let's quickly check to make sure everything's as expected:
import numpy as np
a = np.array([[5, 3, 4, 4, 5, 6],
[27, 2, 3, 4, 12, 6],
[1, 3, 5, 4, 5, 6],
[1, 2, 3, 4, 5, 6],
]).transpose()
a_bar = mean(a)
b = np.array([[5, 4, 3, 2, 1, 3]]).transpose()
b_bar = mean(b)
o_a = std_dev(a, a_bar)
o_b = std_dev(b, b_bar)
ab_cov = cov(a, a_bar, b, b_bar)
assert (a_bar == np.array([4.5, 9., 4., 3.5])).all()
assert (b_bar == np.array([3.])).all()
assert (o_a[3] > 4.1833001 and o_a[3] < 4.1833002)
assert (o_b[0] > 3.162277 and o_b[0] < 3.162278)
assert (ab_cov == np.array([-1., 28., -9., -10.])).all()
print("✔️ OK to continue!")
✔️ OK to continue!
Now that we've got all the building blocks to our correlation function, let's see if we can put everything together and break a single byte of AES. In order to do this, let's take a closer look at what we're trying to do and the data we've got:
print(trace_array)
[[-0.07324219 -0.28833008 -0.16113281 ... 0.11352539 0.12939453 0.1574707 ] [-0.07397461 -0.29248047 -0.16552734 ... 0.12402344 0.13818359 0.16357422] [-0.06542969 -0.27026367 -0.14990234 ... 0.12426758 0.13525391 0.16137695] ... [-0.06469727 -0.27416992 -0.1484375 ... 0.11938477 0.13134766 0.16064453] [-0.06689453 -0.27246094 -0.14916992 ... 0.11572266 0.13037109 0.15795898] [-0.07250977 -0.28955078 -0.16259766 ... 0.12109375 0.13427734 0.16162109]]
You should have something like the following:
[
[point_0, point_1, point_2, ...], # trace 0
[point_0, point_1, point_2, ...], # trace 1
[point_0, point_1, point_2, ...], # trace 2
...
]
where the rows of the array are the different traces we captured and the columns of the array are the different points in those traces. The columns here will be one of the two datasets for our correlation equation. The other dataset will be the hamming weight of the SBox output:
[
[HW[aes_internal(plaintext0[0], key[0])], # trace 0
[HW[aes_internal(plaintext1[0], key[0])], # trace 1
[HW[aes_internal(plaintext2[0], key[0])], # trace 2
...
]
which we'll shorten to:
[
[hw], # trace 1
[hw], # trace 2
[hw], # trace 3
...
]
Like with the DPA attack, we don't know where the encryption is occurring, meaning we have to repeat the correlation calculation for each column in the trace array, with the largest correlation being our best guess for where the SBox output is happening. We obviously also don't know the key (that's the thing we're trying to find!), so we'll also need to repeat the best correlation calculation for each possible value of key[0] (0 to 255). The key with the highest absolute correlation is our best guess for the value of the key byte.
A really nice feature of numpy is that we can do the correlation calculations across the entire trace at once (mean, std_dev, cov). That means there's no need to do:
t_bar = []
for point_num in range(len(trace_array[0])):
t_bar.append(mean(trace_array[:,point_num]))
# and so on...
t_bar = np.array(t_bar)
when we can do
t_bar = mean(trace_array)
and get the same thing back. The only caveat being that we need to make sure that the columns and rows of our arrays are the right way around (i.e. make sure your hamming weight array has 1 column and 50 rows and not the other way around). If you find it easier to construct and array one way and not the other, you can use the .transpose() method to swap the rows and columns.
Once you've got all your correlations for a particular key guess, you want to find the largest absolute correlation. We're taking the absolute value of the correlation here since we only care that the relation between hamming weight and the power trace is linear, not that the slope is positive or negative. max(abs(correlations)) will do that for you.
Perform this for every possible value of the key byte (aka 0 to 255) and the one with the largest correlation is your best guess for the key. It's up to you how you want to extract this information from your loop, but one way of doing it is to stick the best guess for each of your key guesses in an array. Once you've gone through all the key guesses, you can extract the best guess with np.argmax(maxcpa) and the correlation of that guess with max(maxcpa).
from tqdm.notebook import trange
maxcpa = [0] * 256
# we don't need to redo the mean and std dev calculations
# for each key guess
t_bar = mean(trace_array)
o_t = std_dev(trace_array, t_bar)
for kguess in trange(0, 256):
hws = np.array([[HW[aes_internal(textin[0],kguess)] for textin in textin_array]]).transpose()
# ###################
# Add your code here
# ###################
#raise NotImplementedError("Add your code here, and delete this.")
# ###################
# START SOLUTION
# ###################
hws_bar = mean(hws)
o_hws = std_dev(hws, hws_bar)
correlation = cov(trace_array, t_bar, hws, hws_bar)
cpaoutput = correlation/(o_t*o_hws)
maxcpa[kguess] = max(abs(cpaoutput))
guess = np.argmax(maxcpa)
guess_corr = max(maxcpa)
# ###################
# END SOLUTION
# ###################
print("Key guess: ", hex(guess))
print("Correlation: ", guess_corr)
Key guess: 0x2b Correlation: 0.9334191242378663
Let's make sure we've recovered the byte correctly:
assert guess == 0x2b
print("✔️ OK to continue!")
✔️ OK to continue!
To break the rest of the key, simply repeat the attack for the rest of the bytes of the key. Don't forget to update your code from above to use the correct byte of the plaintext!
t_bar = np.sum(trace_array, axis=0)/len(trace_array)
o_t = np.sqrt(np.sum((trace_array - t_bar)**2, axis=0))
cparefs = [0] * 16 #put your key byte guess correlations here
bestguess = [0] * 16 #put your key byte guesses here
for bnum in trange(0, 16):
maxcpa = [0] * 256
for kguess in range(0, 256):
# ###################
# Add your code here
# ###################
#raise NotImplementedError("Add your code here, and delete this.")
# ###################
# START SOLUTION
# ###################
hws = np.array([[HW[aes_internal(textin[bnum],kguess)] for textin in textin_array]]).transpose()
hws_bar = mean(hws)
o_hws = std_dev(hws, hws_bar)
correlation = cov(trace_array, t_bar, hws, hws_bar)
cpaoutput = correlation/(o_t*o_hws)
maxcpa[kguess] = max(abs(cpaoutput))
bestguess[bnum] = np.argmax(maxcpa)
cparefs[bnum] = max(maxcpa)
# ###################
# END SOLUTION
# ###################
print("Best Key Guess: ", end="")
for b in bestguess: print("%02x " % b, end="")
print("\n", cparefs)
Best Key Guess: 2b 7e 15 16 28 ae d2 a6 ab f7 15 88 09 cf 4f 3c [0.9334191242378663, 0.9347091636921677, 0.9158902719872583, 0.91316825741358, 0.8833562724713887, 0.930286867759601, 0.9024276897504525, 0.9413291676301401, 0.9263872839938035, 0.8907532715857314, 0.9276750058325308, 0.9469684227411389, 0.9364134752846959, 0.9639145829985513, 0.9127343962001581, 0.9184216783501178]
With one final check to make sure you've got the correct key:
for bnum in range(16):
assert bestguess[bnum] == key[bnum], \
"Byte {} failed, expected {:02X} got {:02X}".format(bnum, key[bnum], bestguess[bnum])
print("✔️ OK to continue!")
✔️ OK to continue!
We're done! There's actually a lot of room to expand on this attack:
- Currently, the loop needs to go through all the traces before it can return a correlation. This isn't too bad for a short attack, for a much longer one (think 10k+ traces) we won't get any feedback from the attack until it's finished. Also, if we didn't capture enough traces for the attack, the entire analysis calculation needs to be repeated! Instead of using the original correlation equation, we can instead use an equivalent "online" version that can be easily updated with more traces: $$r_{i,j} = \frac{D\sum_{d=1}^{D}h_{d,i}t_{d,j}-\sum_{d=1}^{D}h_{d,i}\sum_{d=1}^{D}t_{d,j}}{\sqrt{((\sum_{d=1}^Dh_{d,i})^2-D\sum_{d=1}^Dh_{d,i}^2)-((\sum_{d=1}^Dt_{d,j})^2-D\sum_{d=1}^Dh_{d,j}^2)}}$$ where
| Equation | Python Variable | Value |
|---|---|---|
| d | tnum | trace number |
| i | kguess | subkey guess |
| j | j index trace point | sample point in trace |
| h | hypint | guess for power consumption |
| t | traces | traces |
- There's a lot more we can learn from the attack other than the key. For example, we could plot how far away the correct key guess is from the top spot (called the partial guessing entropy or PGE) vs. how many traces we used, giving us a better idea of how many traces we needed to actually recover the correct key. We also might want to plot how correlation for a given key guess changes over time.
This "online" correlation equation is the one that the subject of the next tutorial, ChipWhisperer Analyzer, actually uses. It also provides functions and methods for gathering and plotting some interesting statistics.
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