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Ta b l e 1 .
Some variants of the involutions
S
2;1
#
y
1
=
f
1
(
x
1
,
x
2
,
v
)
y
2
=
f
2
(
x
1
,
x
2
,
v
)
x
1
v
⊕
x
2
v
⊕
x
1
x
1
⊕
x
2
⊕
x
2
v
1
2
x
1
v
⊕
x
2
v
⊕
x
2
x
1
⊕
x
2
v
3
x
1
⊕
x
2
v
⊕
x
2
x
1
v
⊕
x
2
4
x
1
⊕
x
2
v
⊕
v
(
x
1
⊕
1
)(
v
⊕
1
)
⊕
x
2
5
x
1
v
⊕
x
2
v
⊕
x
1
x
2
v
⊕
x
1
⊕
x
2
(
1
)(
2
,
9
)(
3
,
17
)(
4
,
25
)(
5
)(
6
,
13
)(
7
,
21
)(
8
,
29
)(
10
)(
11
,
18
)
)
,
I
1
:
(
,
)(
)(
,
)(
,
)(
)(
,
)(
)(
,
)(
)(
12
26
14
15
22
16
30
19
20
27
23
24
31
28
32
(
1
)(
10
)(
19
)(
28
)(
37
)(
46
)(
55
)(
64
)(
56
,
63
)(
47
,
54
,
48
,
62
)
I
2
:
(
,
,
,
,
,
)(
,
,
,
,
,
,
,
)
38
45
39
53
40
61
29
36
30
44
31
52
32
60
(
20
,
27
,
21
,
35
,
22
,
43
,
23
,
51
,
24
,
59
)(
11
,
18
,
12
,
26
,
13
,
34
,
14
,
42
,
15
,
50
,
16
,
58
)
)
.
The difference between the boxes
S
32;96
(Fig. 3c) and
S
−
1
(
2
,
9
,
3
,
17
,
4
,
25
,
5
,
33
,
6
,
41
,
7
,
49
,
8
,
57
32;96
(Fig. 3d) and between
the boxes
S
64;192
(Fig. 3e) and
S
−
1
64;192
(Fig. 3f) consists only in the use of the components
V
1
,
V
2
,...,
V
6
of the controlling vector
V
. It is easy to see that swapping the components
V
j
and
V
s
−
j
+
1
for
j
s
in arbitrary symmetric COS box defines switching between
two mutually inverse boxes
S
n
;
m
and
S
−
1
=
1
,
2
,...,
n
;
m
.
Switchable COS Boxes S
(
V
,
e
)
32;96
and S
(
V
,
e
)
2.2
64;192
Due to symmetric structure of
S
32;96
-box (and
S
64;192
-box) its modifications
S
V
,where
V
,and
S
V
,where
V
=(
are mutually inverse. Such prop-
erty is a core one for the design of the switchable COS boxes
S
(
V
,
e
)
64;192
=(
V
1
,
V
2
,...
V
6
)
V
6
,
V
5
...,
V
1
)
and
S
(
V
,
e
)
32;96
.The
lasts can be constructed using very simple transposition box
P
(
e
)
96;1
implemented as some
single layer CP box consisting of three parallel single-layer boxes
P
(
e
)
2
×
16;1
(Fig. 4a).
Input of each
P
(
e
)
2
×
16;1
-box is divided into 16-bit left and 16-bit right inputs. The box
P
(
e
)
2
16;1
represents 16 parallel
P
(
e
)
2;1
-boxes controlled with the same bit
e
. The right (left)
inputs (outputs) of 16 parallel boxes
P
(
e
)
2;1
×
compose the right (left) 16-bit input (output)
of the box
P
(
e
)
2
16;1
. Thus, each of three boxes
P
(
e
)
16;1
performs
e
-dependent swapping
of the respective pair of the 16-bit components of the controlling vector
V
.
For example,
P
(
0
)
×
2
×
and
P
(
1
)
2
×
16;1
(
V
1
,
V
6
)=(
V
1
,
V
6
)
2
×
16;1
(
V
1
,
V
6
)=(
V
6
,
V
1
)
.Ifthein-
put vector of the box
P
(
e
)
96;1
, then at the output of
P
(
e
)
96;1
we have
V
=
is
(
V
1
,
V
2
,...
V
6
)
0) or
V
=(
(
V
1
,
V
2
,...,
V
6
)
(if
e
=
V
6
,
V
5
,...,
V
1
)
(if
e
=
1). Structure of the switchable
COS box
S
(
V
,
e
)
32;96
is shown in Fig. 4b. In Section 3 we design SCO-based ciphers using
the boxes
S
(
V
,
e
)
32;96
and
S
(
V
,
e
)
32;96
the controlled elements of which are described by pair of
Boolean functions given in rows 1 and 5 of Table 2, correspondingly.
The switchable COS box
S
(
V
,
e
)
64;192
can be constructed with the use of transposition box
P
(
e
)
192;1
that represents three parallel single-layer boxes
P
(
e
)
32;1
(Fig. 4c). Each
P
(
e
)
32;1
-
2
×
2
×
