Cryptography Reference
In-Depth Information
the data is placed in different clusters, each typically covering a range of val-
ues of the attacked intermediate value. Let
F
X
(
x
)and
F
L
(
x
) be the empirical
cumulative distribution functions of the sample populations
X
and
L
.TheK-S
test between the variables
X
and
L
is:
D
KS
(
X
L
)=sup
x
|
F
X
(
x
)
−
F
L
(
x
)
|
.
The CVM test is defined similarly as:
L
)=
x
F
L
(
x
))
2
.
D
CVM
(
X
(
F
X
(
x
)
−
4 Estimators of Mutual Information
Gierlichs et al. in [8] propose the use of mutual information as side-channel dis-
tinguisher in an attack called Mutual Information Analysis (MIA). The authors
present this method as an interesting alternative to the powerful CPA as the
attacker does not have to assume a particular power consumption model for the
targeted device (Sec. 3.1). Indeed, mutual information records both linear and
non-linear relationships between variables while CPA only measures linear ones.
In theory, MIA should be considered more generic as the attacker makes less
assumptions about the device. However in practice, the results of MIA are not
good compared to CPA [29,24,20]. In fact, the eciency of MIA is closely related
to its chosen estimator of mutual information. Some authors studied parametric
estimation methods and their eciency combined with MIA [24,7,16]. On the
contrary, nonparametric estimators are not thoroughly researched [28], although
they fit the original purpose of MIA more suitably.
4.1
Parametric vs Nonparametric Estimation
There are two basic approaches to estimation: parametric and nonparametric.
In this paper, we restrict ourselves to the nonparametric field. Parametric es-
timation makes assumptions about the regression function that describes the
relationship between dependent variables. Therefore, the density function will
assume that the data are from a known family of distributions, such as normal,
and the parameters of the function are then optimized by fitting the model to
the data set. Nonparametric estimation, by contrast, is a statistical method that
has no meaningful associated parameters. There is often no reliable measure
used for the choice of the parameters. However, this type of estimation seems
more suitable to the original purpose of the MIA: a generic side-channel attack
that makes the less assumptions possible. Hence, this paper seeks to introduce
ecient nonparametric pdf estimation methods in the context of side-channel
analysis.
4.2
Histogram-Based Estimator
The most simple and time ecient method to estimate pdf is using histograms.
An histogram consists in a partition of the range of values of each variables into
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