Page 157 - ISCI’2017

      
( )

X ,
b i = θ j K 0 +i   mod ( f ( ) ) ( f 1 ( ) pX , 1 , n ,   m ( ) m       . (10)
p ,
X ,
) f

1

  
   
For conditions (7)–(10) let us present statement 1 for three modulo transformation in form of
theorem 1.
Theorem 1. Deterministic PRS generator, which is functioning according to algorithm of
multimodulo transformations on the basis (1) according to the rules:

      
X ,
b i =   ( ) mod θ j  i  ( f ( ) pX , n , ) ( f, 1 ( ) pX , 1 n , 1 ) f,   m ( ) m       (11)
       

or

      
( )

X ,
) ( f
) f
b i = θ j k 0 +i   mod ( f ( ) p, n , 1 ( ) pX , 1 , n ,   m ( ) mX ,       , (12)

1

      
under fulfillment of conditions (2)–(8) provides generation of PRS (symbols) numbers with undefined
basis of m alphabet, with repetition period p n − 1, with equally possible appearance of symbols at
the repetition period p n − 1 and with ensemble of isomorphism (pϕ n − ) 1 .

Theorem 1 for three modulo transformation proving.
Regarding the last module m it can take arbitrary value and it will be presented as polynomial. Let

us mark that ( ) xf and ( ) xf 1 in (11) are irreducible polynomials, which can be presented over the

field ( ) 2F , i.e. as polynomial of n degree over ( ) 2F .

In regard to repetition period, since { } – primary elements, for providing maximum period
θ
i
n
p n − 1 it is necessary and enough for ( ) xf to be irreducible over the field ( ) [9]. Since ( ) xf
GF
p
is irreducible over the field ( ), according to (1) elements of Galois field are generated with
p
GF
n
period p n − 1 and each element appears only one time.
Let us define m -symbols (finite alphabet) appearance equiprobability degree, i.e. define

conditions, under which symbols of m alphabet appear equally possible. Symbols will be determined

with the help of polynomials ( ) xf m not higher than n degree.
m
GF
n
p
Let us present all elements of field ( ) as positive integers from θ 0 = 1 to p n − 1.
Then let us sort numbers 1÷ p n − 1 according to the ascending order

1 , 2 , 3 ,  f, ( ) f, x ( ) x + , 1 , 2 f ( ), x 2 f ( ) x + , 1 , 3 f ( ), x 3 f ( ) x + , 1  (13)

 p , n − 1− f ( ), px n − f ( ),x  , p n − 1,
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