D:\sbornik\...\c1.DVI


Mathematical Problems of Computer Science 23, 2004, 59{66.

Constr uction of Sequences of N -polynomials Over

Finite Fields of Odd Char acter istics

Ge vo r g M. H a m b a r d z u m ya n

Institute for Informatics and Automation Problems of NAS of RA

e-mail gev@hylink.am

Abstract

In this paper the method of construction of N-polynomials over Fq with q ´ 1
(mod 4) is presented. For a suitably chosen initial N-polynomial f1 (x) 2 Fq [x] of
degree 2 N -polynomials fk (x) 2 Fq [x] of degree 2k are constructed by the iterated
application of following transformation:f (x) ! (2x) d eg(f) f

³
x+´2x¡1

2

´
; ´ 2 Fq; ´ 6= 0.

Refer ences

[1 ] Co h e n S . D ., 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 ls o ve r ¯ n it e ¯ e ld s ,
D esigns, Codes and Cryptography 2 p p . 1 6 9 -1 7 4 , 1 9 9 2 .

[2 ] K yu r e g ya n M. K ., 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 Fq
o f o d d c h a r a c t e r is t ic s , F inite F ields and Their Applications 9 p p . 3 9 -5 8 , 2 0 0 3 .

[3 ] Me yn H ., E xp lic it N-P o lyn o m ia ls o f 2 -P o we r D e g r e e o ve r Fin it e Fie ld s , I., D esigns,
Codes and Cryptography 6 p p . 1 0 7 -1 1 6 , 1 9 9 5 .

N -µ³½Ù³Ý¹³ÙÝ»ñÇ Ñ³çáñ¹³Ï³ÝáõÃÛáõÝÝ»ñÇ Ï³éáõóáõÙÁ Ï»Ýï
µÝáõó·ñÇãÝ»ñáí í»ñç³íáñ ¹³ßï»ñÇ íñ³

¶. Ø. гٵ³ñÓáõÙÛ³Ý

²Ù÷á÷áõÙ

²Ûë Ñá¹í³ÍáõÙ Ý»ñϳ۳óí³Í ¿ N -µ³½Ù³Ý¹³Ù»ñÇ Ï³éáõóÙ³Ý »Õ³Ý³Ï Fq; q ´ 1 mod ( 4 )
í»ñç³íáñ ¹³ßï»ñáõÙ: гٳå³ï³ëË³Ý Ï»ñåáí ÁÝïñí³Í 2 ³ëïÇ׳ÝÇ f1 ( x ) 2 Fq[x]
N -µ³½Ù³Ý¹³ÙÇ Ñ³Ù³ñ 2 k ³ëïÇ׳ÝÇ f1 ( x) 2 Fq[x] N -µ³½Ù³Ý¹³Ù»ñÁ ϳéõóíáõÙ »Ý
f ( x ) ! ( 2 x ) deg(f)f ( x+´2x¡1

2
) ; ´ 2 Fq; ´ 6= 0 Ó¨³÷áËáõÃÛ³Ý ÏÇñ³éáõÙáí:

5 9