Cryptography Reference
In-Depth Information
where 0
j
<
k and l
=
m
/
k . Now, the parity bit is computed for each part, i.e.
A j . Also, the parity is defined for the irreducible polynomial
A j , and denoted by
by excluding the term x m , and the j th parity bit is denoted by
F j ,0
k .
The x module is the main building block in both the bit-serial and bit-parallel PB
multipliers. The following lemma can be presented to obtain the parity prediction
formulation for this module using the multiple-bit parity code.
Lemma 10.5 Let A and A be the input of the x module and its k-bit parity, respec-
tively. The parity of the output of the x module, i.e., A =
F
(
x
)
j
<
A
·
xmodF
(
x
)
, is obtained
by [355]:
A j
a jl 1 + A j +
a m 1 F j .
=
a
+
(
j
+
1
)
l
1
The remaining blocks of the bit-serial and bit-parallel PB multipliers are the XOR
and AND blocks. As in, the single-bit parity approach, the parity prediction in these
blocks are performed using Properties 10.1 and 10.2, respectively.
An alternative approach proposed in [355] is to partition A and F
, which in
fact results in an interlacing parity code and is similar to the approach proposed
in [83].
In [354], a modified multiple-bit parity code-based approach has been proposed
which uses the parities of both operands A and B . In this approach, the partitioning of
the operands is similar to the one explained in [355], i.e., the operands are divided into
kl -bit slices. The parity prediction in the x module is the same as the one outlined in
Lemma 10.5. However, the parity prediction in the XOR and AND blocks should be
modified to incorporate the parity of the operand B . In [354], it has been shown that
using multiple-bit parity for both operands increases the fault detection capability in
comparison to the approach used in [355]. However, this approach also has greater
area overhead in comparison to the one proposed in [355].
In [356], a fault detection approach has been presented for bit-serial (shown in
Fig. 10.1 a using white blocks) and bit-parallel (shown in Fig. 10.2 ) PB multipliers
using
(
x
)
(
,
)
= (
v 0 ,
v 1 ,...,
v n 1 )
and
the code polynomial is a polynomial whose coefficients are the components of V .
A polynomial of degree n
n
m
linear codes. The codeword is defined as V
m is used to generate the code polynomials of degree
n
1 or less and is defined as follows:
x n m
g n m 1 x n m 1
g 2 x 2
G
(
x
) =
+
+···+
+
g 1 x
+
1
,
where G
(
x
)
is known as the generator polynomial and g i
GF
(
2
)
for 1
i
<
n
m .
. Also, let A , B ,
and Q be the encoding of A , B , and Q , respectively. Now, the following properties
can be defined [356].
2 m
Let A , B , and Q be the field elements of GF
(
)
, and b
GF
(
2
)
Q
· A since
Q
· A .
Property 10.3 If Q
=
b
·
A , then
=
b
=
Q
·
G
(
x
) =
b
·
A
.
G
(
x
) =
b
Q
= A
+ B since
Q
Property 10.4 If Q
=
A
+
B , then
=
Q
·
G
(
x
) = (
A
+
B
) ·
) = A
+ B .
(
G
x
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