Cryptography Reference
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where:
f ( u, v )=min
{|
u
v
|
,k
−|
u
v
|}
(7.6)
Function f is introduced to take into account the circular nature of the
addresses. Finally, we call S min thesmallestofthevaluesof S ( j 1 ,j 2 ) , for all
the possible pairs j 1 and j 2 :
S min =min
j 1 ,j 2 {
S ( j 1 ,j 2 )
}
(7.7)
It is proved in [7.19] that an upper bound of S min is:
sup S min = 2 k
(7.8)
This upper bound is only reached in the case of a regular permutation and
with conditions:
P = P 0 = 2 k
(7.9)
and:
P 0
2
k =
mod P 0
(7.10)
Let us now consider a sequence of any weight that, after permutation, can be
written:
k− 1
d ( D )=
a j D j
(7.11)
j =0
where a j can take the binary value 0 (no error) or 1 (one error) and, before
permutation:
k− 1
k− 1
a i D i =
d ( D )=
a Π( j ) D Π( j )
(7.12)
i =0
j =0
We denote j min and j max the j indices corresponding to the first and last non-
null values a j in d ( D ) . Similarly, we define i min and i max for sequence d ( D ) .
Then, the regular permutation satisfying (7.9) and (7.10) guarantees the prop-
erty:
i min ) > 2 k (7.13)
This is because d ( D ) and d ( D ) , both considered between min and max indices,
contain at least 2 bits whose accumulated spatial distance, as defined by (7.5),
( j max
j min )+( i max
is maximum and equal to 2 k . We must now consider two cases:
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