Cryptography Reference
In-Depth Information
Table 11.1
Error detection capabilities of linear and nonlinear codes
Error detection probability
Number of errors
Linear
Robust (
r
is odd)
Robust (
r
is even)
2
k
2
k
−
r
2
k
−
r
0
2
n
2
k
2
n
−
1
2
k
−
1
2
k
−
r
2
n
−
1
2
k
−
1
2
k
−
r
2
k
2
k
−
r
1
−
+
−
+
−
+
−
2
−
r
+
1
2
n
−
1
2
k
−
1
2
n
−
1
2
k
−
1
2
k
+
1
2
k
−
r
+
1
1
−
0
−
−
−
+
2
−
r
+
2
2
k
2
r
1
−
0
0
−
Fig. 11.9
Optimized archi-
tecture, the multiplicative
inverse is split into
t
=
2
separate modules
It is possible to trade off the level of robustness for the amount of hardware
overhead required to transform linear protection to protection based on systematic
robust codes. Instead of taking one multiplicative inverse for all
r
-bit vectors, it
is possible to divide the one large inversion into disjoint smaller inversions while
retaining many of the robust properties outlined earlier. That is, we can replace
multiplicative inverse in
GF
2
r
2
s
(
)
by
ts
-bit disjoint inverses in
GF
(
)
to produce the
nonlinear
r
bit output (
r
=
ts
). Thus, instead of there being two
r
-bit multiplicative
2
r
2
s
inverses in
GF
(
)
for the whole design, there could be 2
t
inverses in
GF
(
)
,asis
presented in Fig.
11.9
for
t
2. Since the number of two input gates to implement the
inverse is proportional to the square of the number of bits at its input, a modification
where
t
=
2 would result in roughly in the 50 % decrease overhead associated with
the architecture based on robust codes. As a consequence this also results in a slight
decrease in the level of robustness and in the introduction of errors which are detected
with different probabilities.
=
