COUNTING IN Zp[x]

International Research Journal of Mathematics, Engineering and IT Vol. 3, Issue 9, September 2016 IF- 3.563 ISSN: (2349-0322)

© Associated Asia Research Foundation (AARF)

Website: www.aarf.asia Email : [email protected] , [email protected]

COUNTING IN

Z_p[x]

1

P. Anuradha Kameswari, 2_{Y. Swathi} Department of Mathematics, Andhra University, Visakhapatnam - 530003, Andhra Pradesh, India.

ABSTRACT

If f(x)Z_p[x] is an irreducible polynomial, the number of polynomials g(x) with

)) ( ( ))

(

(g x deg f x

deg  (g(x),f(x))=1 is the order of the multiplicative group ofZp[x]/(f(x)).

In this paper we propose a formula for this order in the case when f(x) is any primitive

polynomial. We arrive at this formula by introducing analogues _p and_p to Mobius and Euler

functions and defined on Z_p[x], the set of all primitive polynomials in Z_p[x].

Key words : Finite field, Primitive poylnomials

1 Introduction

In the construction of cryptosystems with polynomials in Zp[x] for prime p , the

quotient ring of the polynomial ring in Z_p[x] with an ideal generated by (f(x)), for f(x) a

polynomial in Zp[x] is considered and the group of units of this quotient is taken as the message

space. In this context it is important to compute the order of this group. If f(x) is an irreducible

polynomial then Zp[x]/(f(x)) is a field and the group of units has 1

k

p elements, for

)) ( ( =deg f x

k , [1] [11]. This group of units is given by the set

1} = )) ( ), ( ( )) ( ( < )) ( ( : ] [ ) (

{g x Zp x deg g x deg f x  g x f x , as for any g(x)Zp[x]/(f(x))_,g(x)

is invertible if and only if deg(g(x))<deg(f(x))and gcd(g(x),f(x))=1. [8]

) (x

f any primitive polynomial in Zp[x], we denote this order by p(f(x)) and we prove this

result for f(x) that are not irreducible as well. We develop the formula for _p(f(x)) by using

the analogues _p and _p to Mobius function  and Euler function  respectively.[2][3][9]

We introduce the functions _p and _p on Z_p[x] and prove some results relating _p and_p

in the following section.

2  and  analogues in Zp[x]:

In this section we define two functions _pand_ponZ_p[x] that are analogues to the

arithmetical functions Mobius function (n) and Euler function (n)

2.1 _p an analogue to Mobius function on Z_p[x]:

Definition 2.1.1 A real valued function _p on Z_p[x] is defined as follows :

0. = )) ( ( deg if 1 = )) (

(f x f x

p



If deg(f(x))>0 and = 3... ,

3 2 2 1

1 n

a n a a a

f f f f

f for f_i(x) irreducible polynomials in Zp[x],

   

. o

0

1, = ... = = = , 1) ( = )) (

( 1 2 3

therwise a a

a a if x

f n

n

p



Theorem 2.1.2 For f(x)Z_p[x] with deg(f(x))0 we have

  



1,_0,i_i (₍ (₍ ))₎₎=_>0,_0.

= )) ( (

) ( )|

( _{f deg f x}

x f deg f x

d p x f x

d 

Proof. Let f(x)Z_p[x], then f(x) is a primitive polynomial. If deg(f(x))=0,

0 =

)

(x c andc

f Z_p further note c=1 as f(x) is primitive. therefore

1 = )) ( ( ) ( )| (

x d

p x f x d

If deg(f(x))>0, with ar r a a a f f f f

f 3

3 2 2 1 1

= and D is the set of divisors of f(x)Zp[x] then

for and } factor e irreducibl square no has ) ( and ) ( | ) ( : ) ( { =

1 d x d x f x d x

D } factor e irreducibl square a has ) ( with ) ( | ) ( : ) ( { =

2 d x d x f x d x

D and = )} ( | ) ( : ] [ ) ( { as given

is d x x d x f x D1 D2

D Zp 

)) ( ( = )) ( ( ) ()| ( ) ( ) ( )| ( x d x d _p D x dx f x d p x f x d  





 )) ( ( = 2 1 ) ( ) ( )| ( x d p D D x d x f x d 



  )) ( ( )) ( ( = 2 ) ( ) ( )| ( 1 ) ( ) ( )| ( x d x d _p D x d x f x d p D x d x f x d  





  

now as D₁ consists of the factors

)), ( ) ( ) ( ) ( ( )),... ( ) ( ( )), ( ) ( ( ),..., ( ), ( ), (

1,f₁ x f₂ x f₃ x f₁ x f₂ x f₁ x f₃ x f₁ x f₂ x f₃ x f_r x we

have )) ( ( ) ( )| ( x d p x f x d 



))) ( ).... ( ) ( (( ... ))) ( ) ( (( )) ( ( .... )) ( ( (1)

=p p f1 x  p fr x p f1 x f2 x  p f1 x f2 x fr x

r r r r r 1) ( ... 1) ( 2 1) ( 1 1

= 2 _ 

                       0 = 1) (1 =  r

therefore ( ( ))=0i ( ( ))>0. )

( )|

(x f x p d x f deg f x

d 



2.2 p an Analogue to Euler function  on Zp[x]:

Definition 2.2.1 we define a function _p on Z_p[x] follows:

For any f(x)Zp[x],

1 = )) ( (f x p

 if deg(f(x))=0;

Then p(f(x)) is the number of polynomials g(x)Zp[x] such that

)) ( ( < )) (

(g x deg f x

deg and (g(x),f(x))=1.

Note: ( ( ))= 1

)) ( ), ( (

) (

1 =





n

x f x g

f k x g

i p f x

 wherekf ={g(x)Zp[x]:deg(g(x))<deg(f(x))},

Theorem 2.2.2 For deg(f(x))1 we have

)) ( (

)) ( ( )

( )|

( ( ( )).

= )) (

( _{deg d x}

x f deg

p x f x d p

p p x d x

f 





Proof. Let kf ={g(x)Zp[x]:deg(g(x))<deg(f(x))}.

And note if deg(f(x))=s, then for all g(x)Z_p[x],deg(g(x))s.

we have 1

1 1

0 = )

(x a ax as_xs

g  for a_iZ_p0is

Now as the number of k_f has s

p elements[8]

Possible sequences s

s p

a a a

a , , , , )= ( 0 1 2  1

1 =

)) ( (

1 = )) ( ), ( (

) (





x f x i g

f k x i g p f x 

1 =

1 = )) ( ), ( (

1 =



ps x f x i g

i

=1 ( )|( ( ), ( ))

= ( ( )) ( 2.1.2)

s p

p i d x g_i x f x

d x By theorem



 

)) ( ( =

) ( )| (

) ( )| ( 1 =

x d p

x f x d

x i g x d s p

i







) ( =

)) ( ( )) ( (

1 = )

( )| (

x d p x d deg x f deg p

i x

f x d





1 ) ( = )) ( ( )) ( ( 1 = ) ( )| (





degd x x f deg p i p x f x d x d  )) ( ( )) ( ( ) ( )| ( )). ( (

= _{deg d x}

x f deg p x f x d p p x d 



Therefore _{( ( ))}

)) ( ( ) ( )| ( ( ( )). = )) (

( _{deg d x}

x f deg p x f x d p p p x d x f  



Theorem 2.2.3 For deg(f(x))0 we have

) 1 (1 = )) ( ( _{( ( ))} ) ( )| ( )) ( ( x g deg x f x g x f deg p p p x

f





 where the product runs over the irreducible

factors of f(x).

Proof. If deg(f(x))=0 we have f(x)=c and by definition we have

1 = ) ( = )) (

(f x _p c

p 

 . On the R.H.S the product is empty and as pdeg(f(x)) = p0 =1. Therefore the

result holds for deg(f(x))=0.

Now if deg(f(x))>0 and let er r e e g g g x

f 2

2 1 1 .

= )

( , Then

) 1 (1 = ) 1

(1 _{( ( ))}

1 = )) ( ( ) ( )| ( x i g deg r i x g deg x f x

g p p

 





)) ( ( )) ( 1 ( )) ( ( )) ( ( , )) ( ( .... 1) ( ... . 1 1 1 = _x r g deg x g deg r x j g deg x i g deg j i x i g deg

i p p p p p

    





)) ( ( .. )) ( 1 ( )) ( ( )) ( ( , )) ( ( 1) ( ... 1 1 1 = _x r g deg x g deg r x j g deg x i g deg j i x i g deg

i p p p

       





)) ( )... ( 1 ( )) ( ). ( ( 2 , )) ( ( 1) ( ... 1) ( 1 1 = _x r g x g deg r x j g x i g deg j i x i g deg

i p p p

     





)) ( ( )) ( ( )) ( ( ) ( )| ( ) ( = x f deg x f deg x d deg p x f x d p p p d 



)) ( ( )) ( ( ) ( )| ( )) ( ( ). ( 1

= _{deg d x}

x f deg p x f x d x f deg p p d p 



)) ( ( . 1

) 1 (1 .

= )) ( (

)) ( ( )

( )| ( )) ( (

x g deg x

f x g x f deg p

p p

x

f







where the product is over g(x), irreducible factors of f(x)

Example 2.2.4 Let f(x)=x2x2Z₃[x], then f(x) is an irreducible polynomial

over Z3 and [ ]/( 2) 2

3 x x x

Z is a field with 32 elements, [4] therefore there are (321)=8

invertible elements in this field,that is _p(f(x))=8.

Now by the above formula we have ( ( ))= . (1 1_{( ( ))}) )

( )| ( )) ( (

x g deg x

f x g x f deg p

p p

x

f







) 3

1 .(1 3 = 2))

( 2 2 ₂ 3 x x 



1 3 = 2

8. =

The group of units is given as

1} = )) ( ), ( ( a ,

; =

) ( : )) ( /( ) (

{g x Z3 f x g x axb a bZ3 nd g x f x

Example 2.2.5Let ( )= 22

x x

f and Z₃[x], then ( )= 22

x x

f is reducible over Z₃

and

1) 1)( ( = 2 = )

( 2  

x x x

x

f .

Now by the above formula we have

) 1 (1 .

= )) (

( _{( ( ))}

) ( )| ( )) ( (

x g deg x

f x g x f deg p

p p

x

f







) 3 1 )(1 3 1 (1 3

= 2  

1) 1)(3 (3

=  

2.2 =

Further given f(x)=x22Z₃[x] , 3(f(x))=4 and note that

2} 1,2 ,2 2,2 1, ,

{0,1,2,x x x x x x is the set of all elements of Z3[x]/(x22) and the four

invertible elements are {1,2,x,2x}.

3 Conclusion:

The formula for p(f(x)) gives the order of the multiplicative group Zp[x]/(f(x)) for

) (x

f any primitive polynomial in Z_p[x]; This product formula developed is quite useful in the

construction of cryptosystem with polynomial in Z_p[x]/(f(x)), with the group of units of the

quotient Z_p[x]/(f(x)) as message space.

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