Cryptography Reference
In-Depth Information
Using the coecients a ( l j ,eachofthe m components of the vector d i is
selected or not as the term of an addition with the content of a previous flip-flop
(except in the case of the first flip-flop) to provide the value to be stored in the
following flip-flop. The new content of a flip-flop thus depends on the current
input and on the content of the previous flip-flop. The case of the first flip-flop
has to be considered differently. If all the b j coecients are null, the input is the
result of the sum of the only components selected of d i . In the opposite case,
the contents of the flip-flops selected by the non-null b j coecients are added to
the sum of the components selected of d i . The code thus generated is recursive.
Thus, the succession of states of the register depends on the departure state
and on the succession of data at the input. The components of redundancy r i
are finally produced by summing the content of the flip-flops selected by the
coecients g .
Let us consider some examples.
— The encoder represented in Figure 5.1 is systematic binary, therefore m =1
and c =1 . Moreover, all the a ( l )
j coecients are null except a (1 1 =1 .
This encoder is not recursive since all the coecients b j are null. The
redundancy (or parity) bit is defined by g (1)
0
=1 , g (1)
1
=1 , g (1)
2
=0 and
g (1)
3
=1 .
— In the case of the non-systematic non-recursive binary (here called "clas-
sical") encoder in Figure 5.2, m =1 , c =0 ;amongthe a ( l )
j
,only a (1)
1
=1
is non-null and b j =0
j . Two parity bits come from the encoder and
are defined by g (1)
0
=1 , g (1)
1
=1 , g (1)
2
=0 , g (1)
3
=1 and g (2)
0
=1 , g (2)
1
=0 ,
g (2)
2
=1 , g (2)
3
=1 .
— Figure 5.3 presents a recursive systematic binary encoder ( m =1 , c =1
and a (1 1 =1 ). The coecients of the recursivity loop are then b 1 =1 ,
b 2 =0 , b 3 =1 and those of the redundancy are g (1)
=1 , g (1)
1
=0 , g (1)
2
=1 ,
0
g (1)
3
=1 .
— Figure 5.5 represents a recursive systematic double-binary encoder. The
only coecients that differ from the previous case are the a ( l )
j
:coecients
a (1 1 , a (2 1 , a (2)
and a (2)
3
are equal to 1, the other a ( l )
j
are null.
2
To define an encoder, it is not however necessary to make a graphic repre-
sentation since knowledge of the parameters presented in Figure 5.4 is sucient.
A condensed representation of these parameters is known as generator polyno-
mials. This notation is presented in the following paragraph.
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