Tutorial B8 Profiling Attacks (Manual Template Attack)

Loading the Traces

The traces recorded from the ChipWhisperer Capture program are saved as NumPy arrays. These files can be loaded using the np.load() function. We're interested in the traces, plaintext, and random keys used in our template captures, so we can load this data with the code:

import numpy as np
tempTraces = np.load(r'C:\chipwhisperer\software\temp_attack\rand_key_data\traces\2016.05.24-12.53.15_traces.npy')
tempPText  = np.load(r'C:\chipwhisperer\software\temp_attack\rand_key_data\traces\2016.05.24-12.53.15_textin.npy')
tempKey    = np.load(r'C:\chipwhisperer\software\temp_attack\rand_key_data\traces\2016.05.24-12.53.15_keylist.npy')

(Of course, fill in your own filenames.)

We can check this data to make sure it looks okay. Some useful checks might be:

import matplotlib.pyplot as plt
print tempPText
print len(tempPText)
plt.plot(tempTraces[0])
plt.show()

Get in the habit of checking your data with some basic print statements or plots - it's a good sanity check to make sure that your arrays are the size they should be! If everything looks okay, comment out these checks and move on.

Sorting the Traces

With the data in-hand, our next task is to group the data according to our model. We're attacking an intermediate result in the AES algorithm where

intermediate = sbox[plaintext ^ key]

In our attack, we're going to try to look at a trace and decide what the Hamming weight of this intermediate result is. To set up the templates, we need to sort our template traces into 9 groups. The first group will be all of the traces that have an intermediate Hamming weight of 0. The next group has a Hamming weight of 1, etc..., and the last group has a Hamming weight of 8.

To find which group a trace belongs in, we can calculate the intermediate result from the plaintext and key. Some Python code to do this is:

sbox=(
    0x63,0x7c,0x77,0x7b,0xf2,0x6b,0x6f,0xc5,0x30,0x01,0x67,0x2b,0xfe,0xd7,0xab,0x76,
    0xca,0x82,0xc9,0x7d,0xfa,0x59,0x47,0xf0,0xad,0xd4,0xa2,0xaf,0x9c,0xa4,0x72,0xc0,
    0xb7,0xfd,0x93,0x26,0x36,0x3f,0xf7,0xcc,0x34,0xa5,0xe5,0xf1,0x71,0xd8,0x31,0x15,
    0x04,0xc7,0x23,0xc3,0x18,0x96,0x05,0x9a,0x07,0x12,0x80,0xe2,0xeb,0x27,0xb2,0x75,
    0x09,0x83,0x2c,0x1a,0x1b,0x6e,0x5a,0xa0,0x52,0x3b,0xd6,0xb3,0x29,0xe3,0x2f,0x84,
    0x53,0xd1,0x00,0xed,0x20,0xfc,0xb1,0x5b,0x6a,0xcb,0xbe,0x39,0x4a,0x4c,0x58,0xcf,
    0xd0,0xef,0xaa,0xfb,0x43,0x4d,0x33,0x85,0x45,0xf9,0x02,0x7f,0x50,0x3c,0x9f,0xa8,
    0x51,0xa3,0x40,0x8f,0x92,0x9d,0x38,0xf5,0xbc,0xb6,0xda,0x21,0x10,0xff,0xf3,0xd2,
    0xcd,0x0c,0x13,0xec,0x5f,0x97,0x44,0x17,0xc4,0xa7,0x7e,0x3d,0x64,0x5d,0x19,0x73,
    0x60,0x81,0x4f,0xdc,0x22,0x2a,0x90,0x88,0x46,0xee,0xb8,0x14,0xde,0x5e,0x0b,0xdb,
    0xe0,0x32,0x3a,0x0a,0x49,0x06,0x24,0x5c,0xc2,0xd3,0xac,0x62,0x91,0x95,0xe4,0x79,
    0xe7,0xc8,0x37,0x6d,0x8d,0xd5,0x4e,0xa9,0x6c,0x56,0xf4,0xea,0x65,0x7a,0xae,0x08,
    0xba,0x78,0x25,0x2e,0x1c,0xa6,0xb4,0xc6,0xe8,0xdd,0x74,0x1f,0x4b,0xbd,0x8b,0x8a,
    0x70,0x3e,0xb5,0x66,0x48,0x03,0xf6,0x0e,0x61,0x35,0x57,0xb9,0x86,0xc1,0x1d,0x9e,
    0xe1,0xf8,0x98,0x11,0x69,0xd9,0x8e,0x94,0x9b,0x1e,0x87,0xe9,0xce,0x55,0x28,0xdf,
    0x8c,0xa1,0x89,0x0d,0xbf,0xe6,0x42,0x68,0x41,0x99,0x2d,0x0f,0xb0,0x54,0xbb,0x16) 

intermediate = sbox[tempPText[0][0] ^ tempKey[0][0]]

This will calculate the intermediate value for trace 0, looking only at subkey 0. This calculation can be repeated using list comprehension:

tempSbox = [sbox[tempPText[i][0] ^ tempKey[i][0]] for i in range(len(tempPText))]

Then, we need to get the Hamming weight of all of these intermediate values. We can do this using the same lookup table as the previous tutorial:

hw = [bin(x).count("1") for x in range(256)]
tempHW = [hw[s] for s in tempSbox]

Confirm that these look correct by printing them. (Would I ever steer you wrong?)

With these Hamming weights, we can look at every trace and decide which group it belongs in. Let's make a list of lists of traces:

tempTracesHW = [[] for _ in range(9)]

Then, we can loop through our list of traces and append each trace to the right category:

for i in range(len(tempTraces)):
    HW = tempHW[i]
    tempTracesHW[HW].append(tempTraces[i])

Now, tempTracesHW[y] is a list of all of the traces with an intermediate Hamming weight of y. As our last step, let's turn this into a list of NumPy arrays to make the math easier in the rest of the tutorial:

tempTracesHW = [np.array(tempTracesHW[HW]) for HW in range(9)]

One more check: make sure we have data in any one of these categories by printing

print len(tempTracesHW[8])

Points of Interest

After sorting the traces by their Hamming weights, we need to find an "average trace" for each weight. We can make an array to hold 9 of these averages:

tempMeans = np.zeros((9, len(tempTraces[0])))

Then, we can fill up each of these traces one-by-one. NumPy's average() function makes this easy by including an "axis" input. We can tell NumPy to find the average at each point in time and repeat this for all 9 weights:

for i in range(9):
    tempMeans[i] = np.average(tempTracesHW[i], 0)

Once again, it's a good idea to plot one of these averages to make sure that the average traces look okay:

plt.plot(tempMeans[2])
plt.grid()
plt.show()

We can use these average traces to find points of interest using the sum of differences method. If this sounds unfamiliar, take a look at the theory page for a review of the math. The first step is to create a "trace" that stores these differences:

tempSumDiff = np.zeros(len(tempTraces[0]))

Then, we want to look at all of the pairs of traces, subtract them, and add them to the sum of differences. This is simple to do with 2 loops:

for i in range(9):
    for j in range(i):
        tempSumDiff += np.abs(tempMeans[i] - tempMeans[j])

This sum of differences will have peaks where the average traces showed a lot of variance. Make sure that there are a handful of high peaks that we can use as points of interest:

plt.plot(tempSumDiff)
plt.grid()
plt.show()

Now, the most interesting points of interest are the highest peaks in this sum of differences plot. However, we can't just sort the array and pick the top points. Remember (because you read the theory page, right? ^right?) that we need to make sure our points have some space between them. For instance, it would be a bad idea to pick both 1950 and 1951 as points of interest.

We can use the algorithm from the theory page to pick out some POIs:

• Make an empty list of POIs
• Find the biggest peak in the sum of differences trace and add it to the list of POIs
• Zero out some of the surrounding points
• Repeat until we have enough POIs

Try to code this on your own - it's a good exercise. If you get stuck, here's our implementation:

# Some settings that we can change
numPOIs = 5        # How many POIs do we want?
POIspacing = 5     # How far apart do the POIs have to be?

# Make an empty list of POIs
POIs = []

# Repeat until we have enough POIs
for i in range(numPOIs):
    # Find the biggest peak and add it to the list of POIs
    nextPOI = tempSumDiff.argmax()
    POIs.append(nextPOI)
    
    # Zero out some of the surrounding points
    # Make sure we don't go out of bounds
    poiMin = max(0, nextPOI - POIspacing)
    poiMax = min(nextPOI + POIspacing, len(tempSumDiff))
    for j in range(poiMin, poiMax):
        tempSumDiff[j] = 0

# Make sure they look okay
print POIs

Covariance Matrices

With 5 (or numPOIs) POIs picked out, we can build our multivariate distributions at each point for each Hamming weight. We need to write down two matrices for each weight:

• A mean matrix (1 x numPOIs) which stores the mean at each POI
• A covariance matrix (numPOIs x numPOIs) which stores the variances and covariances between each of the POIs

The mean matrix is easy to set up because we've already found the mean at every point. All we need to do is grab the right points:

meanMatrix = np.zeros((9, numPOIs))
for HW in range(9):
    for i in range(numPOIs):
        meanMatrix[HW][i] = tempMeans[HW][POIs[i]]

The covariance matrix is a bit more complex. We need a way to find the covariance between two 1D arrays. Helpfully, NumPy has the cov(a, b) function, which returns the matrix

np.cov(a, b) = [[cov(a, a), cov(a, b)],
                [cov(b, a), cov(b, b)]]

We can use this to define our own covariance function:

def cov(x, y):
    # Find the covariance between two 1D lists (x and y).
    # Note that var(x) = cov(x, x)
    return np.cov(x, y)[0][1]

As mentioned in the comments, this function can also calculate the variance of an array by passing the same array in for both x and y.

We'll use our function to build the covariance matrices:

covMatrix  = np.zeros((9, numPOIs, numPOIs))
for HW in range(9):
    for i in range(numPOIs):
        for j in range(numPOIs):
            x = tempTracesHW[HW][:,POIs[i]]
            y = tempTracesHW[HW][:,POIs[j]]
            covMatrix[HW,i,j] = cov(x, y)

Make sure these matrices look okay before continuing:

print meanMatrix
print covMatrix[0]

Note that you may not have enough data to complete the covariance matrix - check out the Gotchas at the end of this tutorial if this step blew up on you.

Loading the Traces

If you followed the instructions from the previous tutorial, you'll have a few dozen traces that use random plaintexts and a fixed key. We can load these in exactly the same way as the template traces:

atkTraces = np.load(r'C:\chipwhisperer\software\temp_attack\fixed_key_data\traces\2016.05.24-12.10.07_traces.npy')
atkPText  = np.load(r'C:\chipwhisperer\software\temp_attack\fixed_key_data\traces\2016.05.24-12.10.07_textin.npy')
atkKey    = np.load(r'C:\chipwhisperer\software\temp_attack\fixed_key_data\traces\2016.05.24-12.10.07_knownkey.npy')

In particular, let's check the key to make sure we know what our goal is:

print atkKey

With the default key in ChipWhisperer Capture, this prints

[ 43 126  21  22  40 174 210 166 171 247  21 136   9 207  79  60]

We're only attacking the first subkey, so let's see if we can get 43 to come up as the best guess.

Using the Template

The very last step is to apply our template to these traces. We want to keep a running total of , so we'll make space for our 256 guesses:

P_k = np.zeros(256)

Then, we want to do the following for every attack trace:

• Grab the samples from the points of interest and store them in a list
• For all 256 of the subkey guesses...
  • Figure out which Hamming weight we need, according to our known plaintext and guessed subkey
  • Build a multivariate_normal object using the relevant mean and covariance matrices
  • Calculate the log of the PDF () and add it to the running total
• List the best guesses we've seen so far

Make sure you've included the multivariate stats from SciPy:

from scipy.stats import multivariate_normal

Our implementation is:

for j in range(len(atkTraces)):
    # Grab key points and put them in a matrix
    a = [atkTraces[j][POIs[i]] for i in range(len(POIs))]
    
    # Test each key
    for k in range(256):
        # Find HW coming out of sbox
        HW = hw[sbox[atkPText[j][0] ^ k]]
    
        # Find p_{k,j}
        rv = multivariate_normal(meanMatrix[HW], covMatrix[HW])
        p_kj = rv.pdf(a)
   
        # Add it to running total
        P_k[k] += np.log(p_kj)

    # Print our top 5 results so far
    # Best match on the right
    print P_k.argsort()[-5:]

The output that appears is:

[219 145 181 134 127]
[ 37  43  76 123 235]
[32 44 45 77 43]
[199  45  44  77  43]
[139  37  42  77  43]
[45 77 37 42 43]
[235  37  45  42  43]
[ 37  77 139 235  43]

This means:

• After 1 trace, 43 wasn't even in the top 5 guesses. This is fine - there are probably a lot of key candidates tied for first place.
• After 2 traces, 43 was the 4th best guess - pretty good...
• After 3+ traces, 43 was the best candidate for the subkey.

so we successfully attacked the first subkey in 3 traces! This is a bit lucky - most of the subkeys tend to take a bit more data. Don't be surprised if your attack takes closer to a dozen trials.