Cryptography Reference
In-Depth Information
Fig. 11.3
Percentages of errors detected versus error multiplicities for
a
8-bit linear duplication
b
8-bit robust duplication
satisfy x
1
2
r
=
x
2
, where all the computations are in G F
(
)
. The code is a
2
-robust
2
r
.
code with n
=
2
r and M
=
As an example, Fig.
11.3
shows the percent of detectable errors as a function of
error multiplicity (number of distorted bits) for eight-bit linear and robust duplication
codes
. The detection capability of linear duplication codes depends
largely on the multiplicity and type of the error. The scheme offers relatively poor
protection for errors of even multiplicities, which can be exploited by the attacker to
increase his chance of implementing a successful fault injection attack. In contrast the
robust duplication code has almost completely uniform error detection. This robust
code has
R
(
k
=
r
=
8
)
2. Any error can be masked for at most two messages. Unlike in linear
codes, regardless of which subset of errors is chosen for this robust code the error
masking probability is upper-bounded by 2
−
7
.
Robust duplication codes can be a viable alternative to standard duplication tech-
niques. The application of binary robust duplication codes to memories with self-error
detection and to the comparison between standard and robust duplication techniques
can be found in [213].
The detection of errors for robust codes are message-dependent. If the same error
stays for more than one clock cycle, even if the injected fault manifests as an error
that cannot be detected at the current clock cycle, it is still possible that the error
will be detected at the next clock cycle when a new message arrives. Therefore, the
advantage of robust codes will be more significant for lazy channels where errors have
high probabilities of repeating themselves over several clock cycles. Applications of
robust codes for the protection of lazy channels can be found in [213].
=
11.5.1 Partially Robust Codes
Robust codes generally have higher complexity of encoding and decoding than clas-
sical linear codes. The quadratic systematic codes from Construction 11.1 require
