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FIGURE 3. The classification performance of various machine-learning classifiers: (a) naïve Bayes (purple dashed line), LDA (black dotted
line), and QDA (solid green line), and three support vector machines: (b) a linear classifier (separable), (c) a linear classifier
(nonseparable), and (d) a nonlinear classifier.

vectors denoted as leakage vl for a given value for the secret k,
Pr [ l j | k]. For instance, for the AES example discussed earlier,
256 such distributions would need to be developed, one for
each possible value for KB i (0-255). In fact, one distribution
for each sampling point or the feature for each of the 256 KB i
values is needed. If we continue with three-dimensional
power trace vectors such as [50, 75, 100], we will need to
profile 256 ) 3 = 768 such distributions. Figure 2(c) shows
such distributions for K + 1 different key values.
Bayesian inference allows one to develop a distribution
for the key values, given a specific leakage-a power trace
vector such as [50, 75, 100]. Specifically, the posterior probability that the secret key is equal to k, given any measured
power (l j ), can be computed using Bayes' theorem:
K
Pr [k | l j] = [Pr [ l j | k] Pr [k]] `/ k = 1 Pr [ l j | k] Pr [k] j.
Bayes' theorem derives the likelihood that the given
power leakage l j comes from the key value k as Pr [k | l j]. For
a causal relationship, causes result in effects. The distribution
of effects can be measured by triggering the causes. Bayes'
theorem derives the inverse relationship between effects and
causes; given an effect, it infers which cause is likely to have
triggered it.
Using the maximum-likelihood estimation, the best key
guess is the key value k that leads to the maximum probabiliD
ty k = argmax k ! K % j = 1 Pr [k | l j] . If the prior probability
Pr [k] for k = 1, f, K is uniformly distributed, we get
D
k = argmax k ! K % j = 1 Pr [ l j | k] . The likelihood probability
Pr [ l j | k] determines the kind of classifier. The successful
classifier selects the correct key: k = k *. We next discuss a
variety of classifiers.

SIDE-CHANNEL ANALYSIS USING
MACHINE-LEARNING CLASSIFIERS
In general, there exists a profile of power trace vectors that
were derived from a key value KB i = 100, and there are
Y 100.
power vectors that were derived with key values KB i =
Figure 3(a) shows yellow and blue dots representing these
Y 100 in two-dimensional
two classes KB i = 100 and KB i =
power vectors. The question we need to ask is, given a measured two-dimensional (number of sampling points) power

vector [50, 75], does it belong to the yellow class ^KB i = 100h
Y 100h .
or the blue class ^KB i =
The question gets more complicated when the yellow
and blue dots run into each other, creating an ambiguous
cloud. Machine-learning classifiers are designed to answer
this question or to classify a new data point (e.g., power
vector [50, 75]) into one of the classes. Of course, in general,
there exist more than two classes-in fact, 256 classes for
the AES key block value-and the dimensionality of the
points is much larger than two. Typical power vectors can
have hundreds of sampling points or features, resulting in a
100-plus-dimensional space for classification! A short
description of some machine-learning classifiers follows.
The naïve Bayes classifier is the classifier we described
for the profiling attack. Abstractly, a classifier defines a
boundary to separate two classes. As we see in Figure 3(a),
some points are misclassified (blue points in the mass of
yellow points). Figure 3(a) shows the classification boundary for this classifier. Naïve Bayes is usually computationally expensive.
The LDA classifier defines a boundary between two classes that is a linear function of the feature axes. Figure 3(a)
shows the black dotted line defined by an LDA classifier. In
higher-dimensional spaces, the boundary would be a
hyperplane. For some problems, a large number of points
could be misclassified with a linear boundary. LDA is a poor
match for those problems. LDA is usually one of the most
computationally efficient classifiers, due to linear computation. The quadratic discriminant analysis (QDA) classifier
defines a quadratic boundary as shown in Figure 3(a) as a
solid green line. The classification accuracy depends on the
data set. QDA performs well for instruction-level disassembly classification.
Support vector machines (SVMs) were introduced by Vapnik [13]. They became more important and popular in recent
years when extensions to general nonlinear SVMs were
made. The linear SVM classifier (separable case) defines an
optimal separating hyperplane that maximizes the distance
between the hyperplane and the nearest points on either
side, as shown in Figure 3(b). The linear SVM classifier
JULY 2017

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IEEE Consumer Electronics Magazine

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