D:\sbornik\...\alizadeh2.DVI


Mathematical Problems of Computer Science 35, 26{32, 2011.

Constr ucting M ethods for I r r educible P olynomials

Ma h m o o d A liz a d e h

Islamic Azad University- Ahvaz Branch

E-mail: Alizadeh@iauahvaz.ac.ir

Abstract

In this paper we study the irreducibility of some composite polynomials, constructed
with a polynomial composition method over ¯nite ¯elds. Furthermore, a recurrent
method for constructing families of irreducible polynomials of higher degree from given
irreducible polynomials over ¯nite ¯elds is given.

Refer ences

[1 ] E .R . B e r le ka m p , A lg e b r a ic c o d in g t h e o r y, Mc Gr a w-H ill, N e w Y o r k, 1 9 6 8 .

[2 ] I.F. B la ke , G.S e r o u s s i, N .P .S m a r t , E lliptic curves in cryptography, Ca m b r id g e U n ive r s it y
P r e s s , Ca m b r id g e , r e p r in t e d 2 0 0 0 .

[3 ] J.Ca lm e n t , \ A lg e b r a ic a lg o r it h m s in GF( q) " , D iscrete M ath., vo l. 5 6 , p p . 1 0 1 -1 0 9 , 1 9 8 5 .

[4 ] B .Ch o r , R .R ive s t , \ A kn a p s a c k-t yp e p u b lic ke y c r yp t o s ys t e m b a s e d o n a r it h m e t ic in ¯ n it e
¯ e ld s " , IE E E Trans. Inform. Theory, vo l. 3 4 , p p . 9 0 1 -9 0 9 , 1 9 8 8 .

[5 ] S .D .Co h e n , \ On ir r e d u c ib le p o lyn o m ia ls o f c e r t a in t yp e s in ¯ n it e ¯ e ld s " , P ros. Cambridge
philos. Soc, vo l. 6 6 , p p . 3 3 5 -3 4 4 , 1 9 6 9 .

[6 ] S .D .Co h e n , \ Th e e xp lic it c o n s t r u c t io n o f ir r e d u c ib le p o lyn o m ia l o ve r ¯ n it e ¯ e ld s " , D es.
Codes cryptogr., vo l. 2 , p p . 1 6 9 -1 7 4 , 1 9 9 2 .

[7 ] N .K o b lit z , Algebraic aspects of cryptography, S p r in g e r , B e r lin 1 9 9 8 .

[8 ] R . L id l, H .N ie d e r r e it e r , F inite ¯elds, Ca m b r id g e U n ive r s it y, P r e s s Ca m b r id g e 1 9 8 7 .

[9 ] M. K yu r e g ya n , \ R e c u r r e n t m e t h o d s fo r c o n s t r u c t in g ir r e d u c ib le p o lyn o m ia ls o ve r
GF ( 2 s ) " , F inite ¯elds and their applications, vo l. 8 , p p . 5 2 -6 8 , 2 0 0 2 .

[1 0 ] M.K . K yu r e g ya n , \ It e r a t e d c o n s t r u c t io n s o f ir r e d u c ib le p o lyn o m ia ls o ve r ¯ n it e ¯ e ld s
wit h lin e a r ly in d e p e n d e n t r o o t s " , F inite ¯elds and their applications, vo l. 1 0 , p p . 3 2 3 -
3 4 1 , 2 0 0 4 .

[1 1 ] A .J. Me n e z e s , I. F. B la ke , X .Ga o , R .C.Mu llin , S .A .V a n s t o n e , T.Y a g h o o b ia n , Applica-
tions of ¯nite ¯elds, K lu we r A c a d e m ic p u b lis h e r s , B o s t o n , 1 9 9 3 .

2 6



M. Alizadeh 2 7

âµ»ñíáÕ µ³½Ù³Ý¹³ÙÝ»ñÇ Ï³éáõóÙ³Ý »Õ³Ý³Ï

Ø. ²Éǽ³¹»Ñ

²Ù÷á÷áõÙ

²Ûë ³ß˳ï³ÝùáõÙ Ù»Ýù áõëáõÙݳëÇñáõÙ »Ýù áñáß ÏáÙåá½ÇóÇáÝ µ³½Ù³Ý¹³ÙÝ»ñÇ
ãµ»ñí»ÉÇáõÃÛáõÝÁ, áñáÝù ϳéáõóí³Í »Ý µ³½Ù³ùݹ³Ù³ÛÇÝ ÏáÙåá½ÇóÇáÝ Ù»Ãá¹áí,
í»ñç³íáñ ¹³ßï»ñÇ íñ³: ²í»ÉÇÝ ïñí»É ¿ é»Ïáõñ»Ýï Ù»Ãá¹, í»ñç³íáñ ¹³ßï»ñÇ íñ³
ïñí³Í ãµ»ñíáÕ µ³½Ù³Ý¹³ÙÇó µ³ñÓñ ³ëïÇ׳ÝÇ ãµ»ñíáÕ µ³½Ù³Ý¹³Ù ϳéáõó»Éáõ
ѳٳñ: