Page 158 - ISCI’2017

where ( ) xf is element value of field ( ) x .
f

Let us bring the row (13) according to module ( ) xf 1 , as a result we will get:

1 , 2 , 3 , , f 1 ( ) , -x 1 0 , 1 , 2 , 3 , , f 1 ( ) , -x 1 0 , 1 , 2 , 3 , f , 1 ( ) , -x 1 0 , 1 , ,V , (14)

where 0 ≤V ≤ f 1 ( ) 1.
x
−
Let us present the array (14) as:

   1          
2

, 3 , 2 , 1 , f 1 ( ) x − ; 0 , 1 2,1 , 3 , , f 1 ( ) x − 1 ,0 ;  (15)
    z 1 −          
 
z
,
 ,1 , 2 , 3  , f 1 ( ) x − ; 0 , 1 , 1 ,2 , 3  V ,
;
−
where V ≤ f 1 ( ) 1.
x
On the whole there will be sequence elements z − )1(( f 1 ( ) Vx + in the PRS (15). Besides in the

x
f
last unit there will be no sequence elements beginning with ( +V ) 1 to ( ) 1 and 0.
−
1
−
0 ,
x
Farther on symbols , 21 , 3 , , V appear z times + VV , 1 + , 2  , f 1 ( ) 1 – ( −z ) 1 times.
Probabilities of elements , 21 , 3 , , V appearance will be correspondingly:
z
R 1 = p n − 1 , (16)

V
and + V + , 2  , f 1 ( ) 1
, 1
0 ,
x
−
z − 1
R 2 = p n − 1 . (17)
Thus symbols ,1 , 3 , 2  , f 1 ( ) 1−x 0 , appear almost with almost equal probability, i.e. equally

possible at the period p n − 1 as a result of conversion according to the second ( ) xf 1 module.

Let us observe the stage of conversion according to the third module, which is according to the

theorem 1, can be undefined number ( ) x .
f
m
While analyzing for frequency let us define an array 1,0 , 3 , 2 ,  , f 1 ( ) 1 as
−
x
, 3 , 2 , 1  f , 1 ( ) x . (18)

We will bring (18) together according to module ( ) xf m and get the row:

, 3 , 2 , 1  f , m ( ) x − , 3 , 2 , 1 , 0 , 1  f , m ( ) x − , 3 , 2 , 1 , 0 , 1  f , m ( ) x − , 3 , 2 , 1 , 0 , 1  V , , (19)

where 0 ≤V ≤ f m ( ) 1−x .

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