Cryptography Reference
In-Depth Information
follows. For condition 7 to be satisfied, at least one out of 2d−δ−δ
′
indices
in
2
,
l
l
N −1−
l
2
−(d−δ
′
),N −2−
l
2
− 1 + d−δ
∪
2
must be a filter. Let the number of filters in this interval be s ≥ 1. So the
remaining f−t−t
′
−s many filters must be among the remaining l−2d+δ+δ
′
many indices in
l
2
+ d−δ,l−1
N −1−l,N −2−
l
2
−(d−δ
′
)
∪
.
Corollary 4.7.14. The number of distinct d-favorable (t,t
′
)-bisequences con-
taining at most F ≤ 2l filters is
c(δ,t)c(δ
′
,t
′
)
C
d,F
=
δ≤d,δ
′
≤d
t≤
2
−δ,t
′
≤
2
−δ
′
2d−δ−δ
′
F −t−t
′
−s
2d−δ−δ
′
s
l−2d + δ + δ
′
r
.
s=1
r=0
Proof: In addition to s ≥ 1 filters in
2
,
l
l
N −1−
l
2
−(d−δ
′
),N −2−
l
2
−1 + d−δ
∪
,
2
we need r more filters in
l
2
+ d−δ,l−1
N −1−l,N −2−
l
2
−(d−δ
′
)
∪
,
where t + t
′
+ s + r ≤ F.
Corollary 4.7.15. The number of distinct d-favorable (t,t
′
)-bisequences in
[0,l−1]∪[N −1−l,N −2] (containing at most 2l filters) is
c(δ,t)c(δ
′
,t
′
)(1 −2
−2d+δ+δ
′
)2
l
.
C
d,2l
=
δ≤d,δ
′
≤d
t≤
2
−δ,t
′
≤
2
−δ
′
Proof: Substitute F = 2l in Corollary 4.7.14 and simplify.
Alternatively, the number of distinct sequences satisfying the conditions 1
through 6 is
c(δ,t)c(δ
′
,t
′
)2
l
t,t
′
δ,δ
′
and out of these the number of sequences violating the condition 7 is
c(δ,t)c(δ
′
,t
′
)2
l−2d+δ+δ
′
.
δ,δ
′
t,t
′
