Page 82 - ISCI’2017

Thus, under the implementation of a random substitution s we mean the specific implementation
x
of random vector { ( ), ( ),..., ( )}s y s y 2 s y n and the corresponding value of the ()s y function we
1
x
i
x
x
x
will call the implementation of a random substitution s in the i-th point.
x
s y
Independent random substitution is called such a random permutation { ( ), ( ),..., ( )},
s y s y
x
2
n
1
x
x
for which is true
Ps y Ps y ( ))
y
( ( )) ( ( ))... ( Ps
Ps x ! n x 1 x 2 x n .
( ) =
( ( )) ( ( ))... ( ( ))
∑ Ps y 1 Ps y 2 Ps y n
u
u
u
u= 1
1
−
∀
If iu ∈ , {1,2,..., }: (n P s u ( ))y i = n then
Ps y Ps y ( )) n − n 1
y
( ( )) ( ( )) ... ( Ps
( ) =
Ps x ! n x 1 x 2 x n = ! n = (1)
(( )) (( )) ...(( ))
∑ Ps y 1 Ps y 2 Ps y n ∑ n − n ! n
u
u
u
u= 1 u= 1
and s is called the random, equiprobable and independent (or, in abbreviated form, homogeneous)
x
random substitution.
Thus, the concept of a random homogeneous substitution corresponds to a uniform probabilistic
distribution on the set S = { , ,..., }ss 2 s ! n with the independent implementation of random vectors
1
n
s y
{ ( ), ( ),..., ( )} (2)
s y s y
x
n
2
x
x
1
s ∈ S , x∀∈ {1,2,..., !}: ( )n Ps = ( !)n − 1 .
n
x
x
Modern BSC are commonly described by the random homogeneous substitution model [1, 11],
i.e. it is a standard assumption that probabilistic properties of a processed data blocks bijection
implemented by encryption function, satisfies the characteristics of a random substitution. Indeed, if
random, equiprobable and independent selection of the master key K () x is associated with the choice
of substitution s ∈ S n , then the resulting ciphering transformation will match a random,
x
equiprobable and independent comparison of ciphertext blocks to plaintext blocks on all possible
options of bijective mapping, parameterized by key. For instance, for l-bit cipher with a k bit master
key the model of random substitution will consist of subset
' S = n { ' , ' ,..., ' }s s 2 s 2 k ⊂ S = n { , ,..., }ss 2 s ! n
1
1
l
with random, equiprobable and independent 2 substitutions degree n = 2 (acting on the set
k
Y = { ,yy 2 ,..., y 2 l } of binary data blocks). At that the choice of substitution ' s ∈ x ' S ⊂ n S
1
n
(inplementation of random vector { ( ), ( ),..., (s y s y 2 s y 2 l )}) is set randomly, with equal probability
1
x
x
x
and independently of selected k -bit master key K ()x value.
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