Cryptography Reference
In-Depth Information
4.5
B-Spline Estimator
In [6] Daub et al. introduce the use of B-spline functions as entropy estimators.
A B-spline curve is a generalized Bezier curve. It is specified by the parameters:
-
the degree
d
,ororder
k
=
d
+ 1, so that each segment of the piecewise
polynomial curve has degree
d
or less,
-
a sequence of
m
+1 numbers,
t
0
,...,t
m
, called knot vector, such that
t
i
≤
t
i
+1
,
,
-
control points,
b
0
,...,b
n
.
∀
i
∈{
1
,...,m
−
1
}
A B-spline curve is defined in terms of B-spline basis functions. The
i
-th basis
function of degree
d
,noted
B
i,d
, defined by the knot vector
t
0
,...,t
m
is defined
by the Cox-de Boor recursion formula as follows:
B
i,
0
(
z
)=
1if
t
i
≤
z<t
i
+1
0
other ise.
t
i
t
i
+
d
−
z
−
t
i
+
d
+1
−
z
B
i,d
(
z
)=
B
i,d−
1
(
z
)+
t
i
+1
B
i
+1
,d−
1
(
z
)
,
t
i
t
i
+
d
+1
−
for
i
=0
,...,n
and
d
≥
1. Finally, the property:
n
B
i,d
(
z
)=1
,
i
=0
for any value of
z
, makes B-spline basis functions suitable as a pdf estimator.
This estimator is noted BSE. More details on the use and advantages of BSE in
the side-channel context are available in [28].
5 Experimental Analysis
We analyze in this section the practical eciency of nonparametric estimators
of mutual information in the context of side-channel attacks. We compare their
performances with state-of-the-art proposed side-channel distinguishers:
-
classical parametric test, CPA (Sec. 3.3),
-
nonparametric tests, SPE (Sec. 3.5), CVM (Sec. 3.5),
-
cluster analysis, DCA with variance as criterion function (Sec. 3.4),
-
mutual information with parametric estimation, Cumulant-based Estimator
(CE) [16] which is the most powerful parametric estimator,
-
mutual information with nonparametric estimation, GMIA (Sec. 2.2), HE
(Sec. 4.2), KDE (Sec. 4.3), KNN (Sec. 4.4), BSE (Sec. 4.5).
In order to compare the eciency of side-channel attacks, we use common metrics
proposed in the literature [27]. Guessed Entropy (GE) is the average position of
the correct key hypothesis in the sorted vector of hypothesis at the end of the
attack. Results using another metric are presented in Appendix A.
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