First Principal Components Analysis: A New Side Channel Distinguisher - Information Security and Cryptology-ICISC 2010 - page 412

L

V tot = λ j . (1)

j =1

Given a valid power consumption model and one CS ,therearetwocasesto
be discussed regarding the fluctuation of the total variance when increasing the
number of recorded traces. The first case is the one for which the cryptographic
implementation is not sensitive to the considered CS . In this case, PCA could
not discriminate references of the secret key as well for the other key hypotheses.
The second case happens when the implementation is sensitive to the chosen
CS .Inthiscase, V tot related to V ref k secret key is getting high by increasing the
number of recorded traces. This can be explained by the fact that the secret key
partitioning is the one for which the references are the most different. Intuitively,
for an infinity of traces, V tot converges towards the leakage value. By contrast,
V tot corresponding to one false key approaches the zero value when increasing
the number of traces. This is due to the fact that PCA is not able to discriminate
the references.
In order to highlight the dispersion of the references related to the secret key
with regards to false keys, we carried out an experiment on DES [19] power
consumption traces that are made freely available on line, in the context of the
first version of DPA Contest competition [33]. The DES algorithm that has
been selected for the competition is unprotected and easily breakable by first-
order SCA. More details about this implementation could be found in [11]. For
this purpose, we fixed the ”mean” as CS and the Hamming distance as power
consumption model. Fig. 2 shows the dispersion of references related to the
secret key and one false key, when projected to the first and the second principal
components. These principal components are the most significant given that they
cover a high rate of the total variance so-called explained variance ( EV ). For
the m - th principal component PC m , this rate is defined by the following ratio:

EV ( PC m )= λ m /V tot ,

where λ m is the eigenvalue corresponding to PC m .For m principal components,
we introduce the cumulative explained variance ( CEV ) that is defined by:

m

CEV ( PC 1 ,...,PC m )=( λ i ) /V tot .

i =1

In practice, last principal components are usually considered to be related to the
noise contribution and only few m components are retained for analysis.
The main idea behind using PCA is to reduce the dimensionality of power
consumption traces in order to take account of the secret information for different
time samples and thus to properly exploit the leakage. For this purpose we
used the cumulative variance criteria to extract the significant components. For
instance, we keep only the m first components which explain more than 95% of
the total variance, for each key hypothesis k j .