Cryptography Reference
In-Depth Information
Pearson correlation calculations are based on the assumption that both
X
and
L
values are sampled from a normal distribution. Hence, Pearson rho is part
of parametric tests. On the contrary, methods that do not assume a particular
distribution family for the data are said to be nonparametric.
3.4
Cluster Analysis
Differential Cluster Analysis (DCA) was introduced in [3]. It uses classical cluster
analysis statistics in the side-channel analysis context. The principle of cluster
analysis is to group similar objects into respective categories, i.e. clusters, and
then use a statistical method in order to discover structures in the observed
data. In side-channel analysis, the clusters often correspond to the outputs of the
attacked intermediate value, which is similar to the mutual information technique
presented Sec. 4. Amongst the statistical function used to characterize clusters
proposed in [3], the use of variance seems particularly suited.
3.5
Nonparametric Correlation Statistics
Nonparametric tests make no assumptions about the distribution parameters
of the variables. They do not rely on the estimation of parameters such as the
mean or the standard deviation. Therefore, they are often called parameter-free
or distribution-free methods. The most commonly used nonparametric equiva-
lents to Pearson correlation factor are Spearman R, Kendall tau and coecient
Gamma. The coecient Gamma [9] is similar to Kendall tau and is not very
relevant in our analysis.
The use of the Spearman R has been proposed in [2]. Spearman R assumes
that the variables are on a rank ordered scale. If several values of the variables
are equal, which is the case in the context of side-channel analysis, the formula
for Spearman R is the same as for Pearson's rho. The rank of identical values is
the mean of their respective ranks.
Kendall tau [11] is similar in terms of results to Spearman R. However, its
computation and its statistical meaning is different. Kendall measures the degree
of relationships between variables whereas Pearson and Spearman test the null
hypothesis that there is no relationships between variables. There is different
versions of Kendall statistic. In our context, one should use the coecient that
makes adjustments for tied values:
N
c
−
N
d
τ
b
=
(
N
(
N
,
−
1)
/
2
−
t
)(
N
(
N
−
1)
/
2
−
u
)
where
N
c
is the number of pairs ranked in the same order on both variables,
N
d
is the number of pairs ranked differently on the variables,
t
is the number of tied
values in the first variable,
u
is the number of tied values in the second and
N
is the number of observations.
In [29], the authors propose to use other nonparametric statistics: the
Kolmogorov-Smirnov (K-S) test and the Cramer-von Mises (CVM) test. These
tests are very similar to the DCA and the mutual information analysis. Indeed,
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