Tópicos de Corpos Finitos com Aplicações em Criptografia e Teoria de Códigos

26- Colóquio Brasileiro de Matemática

Tópicos de Corpos Finitos c o m Aplicações em Criptografia e Teoria de Códigos Ariane M. Masuda Daniel Panario

impa

ISBN 978-85-244-0261-6

'"7 8 8 5 2 4

402616

publicações matemáticas no impa

As Publicações Matemáticas

divulgam

trabalhos expositórios que tanto podem resultar de pesquisas como de cursos, seminários ou reuniões científicas patrocinadas pelo IMPA. Em especial, os textos dos tradicionais Colóquios Brasileiros

de Matemática

virão a

público dentro desta coleção, ressaltando assim a postura integrada adotada pela Instituição com relação às atividades de ensino, pesquisa e divulgação da Matemática. Para outras informações, consulte: http://www.impa.br

impa

instituto nacional de matemática pura e aplicada

Tópicos de Corpos Finitos com Aplicações em Criptografia e Teoria de Códigos

Publicações Matemáticas

Tópicos de Corpos Finitos com Aplicações em Criptografia e Teoria de Códigos Ariane M. Masuda University of Ottawa Daniel Panario Carleton University

ílí^Pf^ 26^ Colóquio Brasileiro de Matemática

Copyright © 2007 by Ariane Masuda e Daniel Panario Direitos reservados, 2007 pela Associação Instituto Nacional de Matemática Pura e Aplicada - IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ Impresso no Brasil / Printed in Brazil Capa: Noni Geiger / Sérgio R. Vaz

26"* Colóquio Brasileiro de Matemática • • • • • • • • • • • •

Aspectos Ergódicos da Teoria dos N ú m e r o s - Alexander Arbieto, Carlos Matheus e Carlos Gustavo Moreira Componentes Irredutíveis dos Espaços de Folheações - Alcides Lins Neto Elliptic Regularity and Free Boundary Problems: an Introduction Eduardo V . Teixeira Hiperbolicidade, Estabilidade e Caos em Dimensão Um - Flávio Abdenur e Luiz Felipe Nobili França Introduction to Generalized Complex Geometry - G i l R. Cavalcanti Introduction to Tropical Geometry - Grigory Mikhalkin Introdução aos Algoritmos Randomizados - Celina de Figueiredo, Guilherme da Fonseca, Manoel Lemos e Vinícius de Sá Mathematical Aspects o f Quantum Field Theory - Edson de Faria and Welington de Melo Métodos Estatísticos Não-Paramétricos e suas Aplicações - Aluísio Pinheiro e Hildete P. Pinheiro Moduli Spaces of Curves - Enrico Arbarello Noções de Informação Quântica - Marcelo O. Terra Cunha Three Dimensional Flows - Vítor Araújo e Maria José Pacifico

•

Tópicos de Corpos Finitos com Aplicações em Criptografia e Teoria de Códigos - Ariane Masuda e Daniel Panario

•

Tópicos Introdutórios à Análise Complexa Aplicada - André Nachbin e Ailín Ruiz de Zarate

• •

Uma Introdução à Mecânica Celeste - Sérgio B. Volchan Uma Introdução à Teoria Económica dos Jogos - Humberto Bortolossi, Gilmar Garbugio e Brígida Sartini Uma Introdução aos Sistemas Dinâmicos via Frações Contínuas - Lorenzo J. Diaz e Danielle de Rezende Jorge

•

ISBN: 978-85-244-0261-6

Distribuição: IMPA Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, RJ E-mail: [email protected] http://www.impa.br

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 α∈M  ­ E"¬3 , G/9

 6/9

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 6/9 /

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-i-.E FX>1 S m1 ³ >1W61hE tW-y C 6
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0/9t
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. ³ >1W61hE tX>1¥B9 D
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,E| VX>1 grau(r2 −r1 ) < grau(g) h1 −h2 = 0 r2 −r1 = SAg   S 0 h 1 = h 2 r1 = r 2  X>1

HL$W< =61© E=$/1LE6-s -.>fE

³  ­ 

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R

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² :<®,E

6

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M Hp61, , . - cX>1p« C;43 C 6
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p/9

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S

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i

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F d ∈ F [x]


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h ∈ F [x]

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r1 f g r1

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r2

g

3 -

= q1 g + r1 = q 2 r1 + r 2 =S q3 r2 + r3

 6/9

q1 , . . . , qs+1 , r1 , . . . , rs

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S S

rs−3 rs−2 rs−1

r

1 f C 6
.=6;>6/9„ C Ez, D
 1-.9 C, .  M

= qs−1 rs−2 + rs−1 = qs rs−1 + rs = qs+1 rs + 0, .~W< =61© E= O D ¬1-

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grau(g) > grau(r1 ) > grau(r2 ) > · · · > grau(rs−1 ) > grau(rs ) ≥ 0.

q  E|P

grau(g)

HA43617
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 1<=[  }~v/9

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Hv« C;C4W C 6
„/9”
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a−1

H|-.,<= §, /1f-.vX>1

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f (x) = x3 + 3

g(x) = 2x2 + 2

 6/9

f, g ∈ Z5 [x]


S

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x3 + 3 = 3x(2x2 + 2) + (4x + 3) 2x2 + 2 = (3x + 4)(4x + 3) + 0. jb6
i~

mdc(x3 + 3, 2x2 + 2) = (4−1 )(4x + 3) = 4(4x + 3) = x + 2

 S0³  ­ E

6 f (x) S =
,2x + x3 + x2 + 2 E| OX>1 f, g ∈ Z3 [x]



g(x) = x4 + x2 + 2x

S

 6/9

2x6 + x3 + x2 + 2 = (2x2 + 1)(x4 + x2 + 2x) + (x + 2) x4 + x2 + 2x = (x3 + x2 + 2x + 1)(x + 2) + 1 x + 2 = (x + 2)(1) + 0. jb6
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mdc(f, g) = 1 mdc(f, g) = 1

S

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f

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f



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mdc(f, g) mdc(f, g) = af + bg js D
.”-,1-., 6
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F [x]

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q E,},6/1Y/9$¬3=B;Y-i\ C Ef M W<[\W6;w1<=
 E^CB91-., . .~ M ¬9
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rs rs−1

= rs−2 − qs rs−1 =S rs−3 − qs−1 rs−2 S S

r3 r2 r1

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= r 1 − q 3 r2 = g − q 2 r1 = f − q1 g.

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s=3

S q  E$¬1 6E A CB91-., . žI

= r1 − q3 r2 = r1 − q3 (g − q2 r1 ) = r1 (1 + q2 q3 ) − q3 g = (f − q1 g)(1 + q2 q3 ) − q3 g = f (1 + q2 q3 ) + g(−q1 − q1 q2 q3 − q3 ). a = 1+q q



b = −q − q q q − q

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.-.-* *W< =61© E=

 -67

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g

a



b

S

 :7 7 7 ³  ­ E 7 f (x) = 2x10 + C xE| + x2 + 1 g(x) = x7 + 1 ^ CE¥¬1 6},~f< =61I-P/1 S R E P, . C-.:,-0 P Z3,[x] mdc(f, g) f <=f<= -.7
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., *CB91-., . žI ° S `8

2x10 + x7 + x2 + 1

=

(2x3 + 1)(x7 + 1) + (x3 + x2 )

x +1

=

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x3 + x 2

=

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7

2

2x + 1 g  

=

mdc(f, g) = x + 1

(2x + 1)(x + 1) + 0. S +

<=H,E{/9[ . D M
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x+1

+

=

(x3 + x2 ) − (2x + 2)(2x2 + 1)

=

(x3 + x2 ) − (2x + 2)((x7 + 1) − (x4 + 2x3 + x2 + 2x + 1)(x3 + x2 ))

=

(2x5 )(x3 + x2 ) + (x + 1)(x7 + 1)

=

(2x5 )((2x10 + x7 + x2 + 1) − (2x3 + 1)(x7 + 1)) + (x + 1)(x7 + 1)

=

2x5 (2x10 + x7 + x2 + 1) + (2x8 + x5 + x + 1)(x7 + 1).

. D E

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mdc(f, g) = (2x5 )f + (2x8 + x5 + x + 1)g

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[/1O61I 
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.6
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.\E =Ef<=[/1/1$/9 S q E M grau(r) < grau(f ) r = g − fh I -i>v/9  E1<=[ X>1 y . =E S ` -
.6
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.~ M E * CE ˆ ˆ F f |f f |f f fˆ E©61[ C M ¬1
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F = R[x]/(x2 + 1) = {r0 + r1 α : r0 , r1 ∈ R

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α2 + 1 = 0}.

H|=-.-.,/9>9
.²=: < D ¬1-.

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grau(f ) > 2 f= (x2 + 1)h + r  6/9 S + . =E S q E 2 ¯ f SA =g r+ (x + 1) r= r + r x ∈ R[x] x2 + 1 ∈ ,E M D,0>1PX1>1 M 6/1 2 2 ¯ ¯ f = r +r α (x + 1) x ++ 1 O= 0 F α=x ¯ xW-
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f + g = f¯ + g¯

f g = f¯g¯.

mo *B91<=[ C=
.E6
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q  E|

(a + bα) + (c + dα) = (a + c) + (b + d)α (a + bα)(c + dα) = (ac − bd) + (ad + bc)α.

F

 6;:-i */1

H0>1E( C -W M xW-.>16
.y6
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r

6~0 .~OE¥¬3 6;>1< 


+ C 6
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C 




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mdc(f¯, α2 + f¯g¯ + (α2 + (a+bα)−1 = F = R(i) =

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F

F [x]/(f )

{a0 + a1 α + · · · + an−1 αn−1 : a0 , a1 , . . . , an−1 ∈ F

Œ

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f (α) = 0}.

R ,-,E| 2>1E(
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63 ~|/9y -.3 S

6

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. f ∈ F [x] ~L< H,¬1-.  V ¬- M /9=§,E K F

D¥CB9 D
¥>1E

W< =61© E=

D ­ ¥<= H¬1-.[ C¥ ¬-

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M ,E
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K F

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[K : F ]  K = F (θ)

grau(M )

Ž‹I‹ŽŒ

 ƒ=‰‘¡Ž‹[iŠ,#

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¤LC4361=E x>1E¶hEE-43 Ev/9


S

S q  E|

,E| sX>1

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M X>1L6
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F HL>1E(1-.9/9K f ∈ F-[x] >9
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polinˆomio

elimina¸ca˜o de fatores repetidos

ERF

parte livre de quadrados (cada fator aparece uma vez)

fatora¸ca˜o em graus distintos

outra parte (fatores repetidos)

DDF ...

polinˆomio contendo ... polinˆomio contendo todos os fatores de grau bn/2c todos os fatores de grau 1

fatora¸ca˜o de graus iguais

EDF ... fatores irredut´ıveis de grau 1

...

EDF ...

... fatores irredut´ıveis de grau bn/2c

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x16 + x15 + x14 + 5x13 + x12 + 18x11 + 4x10 + 12x9 + +11x8 + 12x6 + 16x5 + 5x4 + 2x3 + 2x2 + 3x + 1 ∈ F19 [x]

elimina¸ca˜o de fatores repetidos

ERF

x8 + x6 + 2x5 + 18x4 + 17x + 18 x8 + x7 + 2x5 + 17x3 + 18x + 18

fatora¸ca˜o em graus distintos

DDF

x2 + 18

fatora¸ca˜o de graus iguais

EDF

(x − 1) d8= >1-.› S ›1°

x6 + x5 + x4 + 3x3 + x2 + x + 1

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¬ , /96f ih–: Gw1¬1<  , S d8=B1E

/9= 7
i< S

EH
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f1-. E|=?

g63/9  a 0 P <1a-.;< p . D− 1/9  ,    O 0 S U (g, g a (mod p), p) 6 - 9
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. *¬7B9 S


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S +

p

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<=[ CyI  < h1y>1Eff ih–: x1-.=:/

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a 0
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-.=

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S

r 1 < r < p−1 S £ C
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eS 1 ≡ g r (mod p) EK (M ) = (e1 , e2 ) 6 "!  DK (M )(e1 , e2 ) ≡ e−a 1 e2 (mod p)

`A-i¥:-.=43 -E 2/9, -  }~ M ¬ -.:E 2X>1

DK (M )(e1 , e2 ) ≡ (e1 )−a e2 ≡ g −ar e2 ≡ M br b−r ≡ M

 

*



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³ ,},~

 



  




(mod p).

1254     

2 

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iE f¬-,:E,6
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B
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iEf61h S +

. D E( 6/9 M ¥  >1-i6},y/9<=  -=
E$/9[ . C-.C
|E¨CB;
,69?

ž;, ¬1 6-.[ 8, D
.t C E1- EC
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t  >1-.6}

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f C E¨ C -W 21-. E| S

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g F∗q h ∈ F∗ #Q†@…ƒ CŒ 
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E|/1 . C-.C
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i C 63< E|,6
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 /9x>E{/1 /9< E6


h ∈ F∗

S

q =
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,-I  .6
., y .~ o1-i
. , M  | C -W y461

u  -i< M

F

n

M 6/9

p

Ht>1E{1- E3IX >61

n

Hx -.6/9 S

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D,p>1-i6},y/9LE$>1=
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. . D[ 
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6/974W C>1<[/1/1c/9c-I D <=: -f- ¬1<=,Ef«/9< -.7
.E|

/1 . C-.C
. S

.0E

–:63}C$ E3 -D
i6
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i|-I$ G/1:x¥/9I D,6: < :;=E6
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   ƒ 9ƒ=Š\Ž Œ ˆWŽ‹.ŒC M C-.  /9$W- œ  , D
-.6\tmp <=< -G—=I›› ˜ S jbE( > /1 M

E D
-i-.E Q ELI 
.*E|H
9/9t]_>163 C 6G*6
.~L: -.E >G6<=[ D M X>1t]_  -  - E|,6
PI 
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S S car(Fq )

i

i

      

  

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>Eµ 6 ­ >6
.”/9v3 <= 61© E|  $=-.-.,/9>9
.²=: [ y ¬1-. ¤P  E X>1„
9/9  W< =61© E= E

S

F

M 6/1

p =


±,E p -.>E61 -

i

i

i

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i    
   xª 

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>^= ><5

m

S jF 
i-.E 2 6
.-., .  /9 s61 *W<  61©E  *X>1G]
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C E|<=
.E,6
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m

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[ G3 <= 61© E|  G .~„ C6h1,  /9 L E






>1E

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∗S

F ∼ F = F [x]/(f ) g |,<=,E6
\q > ­ <=p  -.=
E|X>1,-,E * < >1< - S q

+ < HE

EHC
9/9„/9v <[ C><=v/9¥² 6/9[ CI t C63 D[ 
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I °P^ C 6 ? U


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E 2 6/9 :; /9>3[ S

3  '



js . C <=h„>E

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,=-.

M < ,
.-. E6


M

s 1 ≤ s ≤ q−1 .
. D]§,6/1 h h ≡ gs E ]
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.E,6
. S^R -.743X>1^ D (mod f) grau(h) < n h C E|¥>1E(1-.9/9>9
.¥/9t --.,/9>1
² :[ FE G-,17
iy Sb³ G6~ M /9, . -D
. S s 1-.; , . DI  < h16/1f>1E 61: 6
. - 
Hy,6 C 6
.-.X>1x]
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jbEz  >1[/1 M /9C
.-.E=61

v∈S

jb6
.~ M .< :t¥IX>},~

s≡

X

ev (h) logg v (

E9/

v∈S

£ 7
ic  <[ C><= | C Efc
H„¬9
.-$D>1E

q − 1). W> cEf „X>1

-.>1±63 C[ S ¤LW[ b-., < :–ty [ 
.Efx-.x/9
,-E 6S

3  '

v∈S



js . C <=h3Y>1E

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,=-.

M

logg v

#S

69?

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9/9

M < ,
.-. E6


s 1
i≤ s ≤ q−1 <X>1 C E h h ≡ h∗ g s (mod f ) t]
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.E6
x,E¨ --.,/9>1
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/9OEf61,=-iG>161=]_-.E S jb6
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grau(h) 1E D ¬1-. SG³ $6~ M /9, . -D
. S s =6
. - Dt]
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Y

61: 

v ev (h) ,

v∈S


,E| OX>1tf<=  -.=
E¥/9[ . C-.C
f/9I D ­ /9H

X

v∈S

i

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ev (h) logg v

(mod q − 1).

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D,-L,E|-, /1 L-iY ,<[ C>1<[-x L
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., b/1, . DI b < >9?

E,/1 -[ S U

<  S .PE0
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E Q/9 . C-.C
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¬ O/90/1/1 HO6
i~t> ./1G61t`A . D

-.t < >1<[

< -.7
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. 2-D
. >1< -I S


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6< [ DY/1pE|H
9/9”/1” < >1<=o/1\²=63/9 , $-IX >-yoI 
.>/9”/p1-.?

¬¬1 <=[/1/1s/9bX>1b>1ENW< =61© E=0H

m

?@ D>3–: S ¤LA]
 M E

E¥¬3  5 D?

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-. - M
.EfE LW<  61©E  t< ,
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H¬9
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3 <= 61©E 

m

+

?@ D>–:  S

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M , D
.

1-.¬3¬1 <=[/1 /9G/-.  O 6/1 }žI 2-i|-i-2\<  -=
E S +

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.=Ef
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/1, D
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i

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.«E61 -i S

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.E |>1Efc-,1-I D,6
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-.E |/9v>1Efo]_>163}~c,-. /9-i S



q > ih&+| E

> .63/9$x] -E¥>1<[y/9 y E¨fEH
9/9Y/9

jb6
.~ M

EHC
.;/1 s/1L 6
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i/9 O . D 6

[ C S /9<6; + §¦ G,E¨—   –˜ *
,E [/9t> ?

js D
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.-.
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>1
-i Q:–-[6
.,

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i|, D
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H,[¥  >1E¨|EI DE, .X>1Efv— ›  M ¢  M ¢› ˜ S q  6
.=6;>-,E P E¶v63<  |/1 E  OE©6  0/9$-i> W< =6©E= O D ¬1-.

Fq

H

n

¬-

P (z) =

Fq

X

k≥0

H

U

q

/1< +;§¦  S|U

+ nS

qk z k =

  E

6wE|,-„/9W< =61©?

M |]_>16},~f,-. /9-i|/9

1 . 1 − qz

.PEœ >9
-iœE61 -ic/9o 6
.-œ6;w1E,-e/9pW< =61© E= | ¬-

F ]
 -I V -.-I/9>9
² :, S8³ q ­ 


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Ik

Y6;w1E-.Y/9¥3 <= 61©E  PE|© 61[ C P=-.-.,/9>9
.²=: [ P ¬1-.

.PE¨6;>1E,-. /9-O/9t>1E¨]
 -O=-.-.,/9>9
.²=: <®41B9f/9x -.>

1 + z k + z 2k + · · · ,

61| D,6
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² :,<-I C*6y/1, C E3 D[}~t/10>1E

i

i

Fq k

/9$ -.>

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H

k

S

X >P, D
L3 <= 61© E| 

3 <= 61©E  S .P .6/1y -.

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3 <= 61© E|  * ¬1-.

P (z) =

Fq

Y

k

1+z +z

2k

k≥1

q 6 <=>²=E *X>1

P (z) =

Y

k≥1

³  ­ 

>œX>1e

]_>6}~Z -i/9 -.«/9

39/9t
iE¥¬3H,E{ -*, . C-.7
i CE

+···

1 1 − zk


Ik

 Ik

Y

=

k≥1

=

1 1 − zk

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.

1 . 1 − qz

L6;w1E-.L/923 <= 61© E|  QE©61[ C Q ¬-

/1F -.>

Nq (m; n) F n ?n >–:I S .P .6/9$y-.>1E6
.y6
,- - M /9,:E bq C 6 D[/9,-.m ]
.-., 2=-.-.,/9>9
.²=: [ */9x-i>v
H Sbu I 
.x   M ¥]_>163}~ -i/9 -.$H m X>1P .~

Sm (z) =

X

n

Nq (m; n)z =

n≥0

k=1

6wE|,-0/1,  ­ /1P/9bW< =6©E= U

m  Y

m

?n >–: , ®E

1 1 − zk F [x]

 Ik

.

HbL C;C43 =,6
s/9

E Ht Cq 6 
i6
 M $B
.-. }~ M /9,61
i/9|3 Ss³  n z/9nP 43S Cm (z) [z ]S m m (z)  6
.sHF  E|<=I M > .6/10PEHC
.9/9P/1L6<=[ DF/9F =6>1<[-. //9I /9, 6;: < :[/9pW-d8<  ­ < C
¥ U

/9<6+;§¦—   M  ,˜ Seu œ6
.6
. M -.o

1< [ },~o/9”E|H
9/9œ/9c ,<[ C>1< ”/9^²=6/1 , M 1-I C[ E y/9 X>6/1 U

x[  | E1<  , C63 D[/9-i–:,<=E,6
.Lf6<   S

n→∞

M >161=]_-.E|,E6
x-.

Nq (m; n)

M

¬ -.:–E AX>1P bB;W66
I s .~$-¬1=
-i-.  P/9P]
.y s-I D>1<=
. /9 +

nδ ≤ m ≤ n1−δ





δ>0

1-:¥¬ [?@ L6¥] -E¥>1< /9

N (m; n) = =

i

m

61f 6
-.:< 

m → ∞

n1/100 ≤ m ≤ n99/100 .

~¥:<=[/9 *3-i

i



/9< +;§,¦v—= –˜ 6  >1=>v>1Ef|, D
 Ef
 :f . D 6

[ |/9

X>6/1

U

n→∞

S

q > ih&+

Z 1 Sm (z)z −n−1 dz 2πi |z|=r Z π
1 Sm reiθ r−n e−inθ dθ, 2π −π

i

i

i

i

i    
   xª 

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-.„>E











  
  
 


6;w1E-.p-.,<

M

r 0


/9<6+;§¦pI 
.=E>eI 
ip=6
.-i<

SoU

  =6
.
[ E6
O
.-.–: H, /9O>1Efx1<=[  }~L> ></9xEHC
.;/1y/9x3 6
 /9y <[ S + 1-I D,6
.-.E Ff/9,- :},~/9< t-i|>1E 1-.:$ -  6<®/1 ³  ­ 

/9< +;§,¦|Hx-i U

r0 = r0 (m, n)

'1: 

F

-i< M E¥¬3 -.

q

­ ^—=  M +

3±,6/9[ C

+ ˜ S

|w161[ f < >}2~|/1|IX >3}~ 0 rSm (r) = n. Sm (r)

7L>6/9 n → 1−δ  M
,E

n

∞

 \>617]_ -E,E|,6
\-.

b(r) = /9<6+;§¦ t
.E¥¬3H,E

E 
.- >X>1

W-

r0 =

1 exp q





0 Sm (r) Sm (r)

0

.

r0 = r0 (m, n)

HP  =6
.
[ E6
O/1 /9

log(n/m) + log log(n/m) + o(1) m

,6
i~61 6
.-.:< |/1 /9¬9
.E OX>1

1 r0 − = O q +

nδ ≤ m ≤

Sm (r0 )r0−n , N (m; n) ∼ p 2πb(r0 )

 6/9

U

61p 6
-.:< 

m

<=H,Ez/9 .  M < $E D
-.>”X>1


i/9 M 6 <=>1² E OX>1



log n nδ



b(r0 ) ∼ 4mn

,

. S

q E¥¬1 66/1\, . DI 0-., >1<=?

n

Nq (m; n) = q n e−(1+o(1)) m ` 9/9E x C,<=,-.-L„`A . 



 M >E|,6
.63/9

n log( m )

m

S

.

q  6
.>/9 M  . ^ E|?

1< [ -.[o6;>1E -Ef§,6E6
oEf= -|/9„¬ Dv/9p/1/1 ¥^>1E /9|CB9I C>},~\Ef -t/9v`A . Do S$U <= -.7
.E| MU

i

i

/9< +;§,¦|¬9
.HE

M -.


EW


 E| §,6/9^
9/9 P P-
EC
-. L/9 M E< h1-O:<= -23-i

q=2 s n log n . m= 2 log 2

m

°

i

i

i

i

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   xªš

i

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dQ 6<=E6
. M o> .E

L(n) = e(1+o(1))

√ n log n M

6
i~««
.E3

/9\B;I C>},~p/9”<= -.7
.Ev¬3   M X>1^ <[ C>< „ y² 6/9[ C, M H\<= E7
i/9 M 6/1

3 -

√

Ho>1Ef« C 6 D
.6
.o <[ C>< –: <0X>1 M 3-i

L(n)c c S 2 log 2 = 1.1774 . . .

 



 





 



 < ¦AC
< S —   M  –˜ 1-.> D,-.E

M H

   





q = 2

>1EfO:-[6
F/90EHC
.;/1L/9P ,<[ C>1< P/9


|…†nŒ‚Tƒ=ŠIŠ S

²=63/9 , M  -iO C6h1,  /9G C E|P…ƒ ! Š‚T‹ˆ† ‘fŠyŽ Œ

js D
.L:–-[6
.

E< h1-i**
,E|WL/1bCB9, >}~O/90EHC
.;/1 M =6
.-9/9>1§,=63/9O>1E¯-.>1E,6
. h1>-²[ 
. 0X>1A]§bP6<=[ DA63~0 -5-.= -. . SVU



-i/9E

E< h1-iE6
.263~OHs/9

q  6
>/9 M  C< -ivo<  -=
E„ C63 D[/9-i–:,<=E,6
.

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. , S

6|1-i
[  M CE{>1E{>1E6
,6
.-x/9>3 *x/9§x: §,, ,':  ­ v—  ˜ S <= -.7
.E||Hy =E < -PŸ|:-i .~|¬  ,1-., 6
. /1¥6f DI}~6
C? U

- - S jbB;,E1<=43 -,E btE¥>/163} M > .6/9¥W6 Ay D,>163/9x, D
.  /9L<= -.7
.E M [ D
PH M L ,<[ C>1< P/9 V< -.7
.E|  6/9 :[/9> M >1EfP: §sX>1 < >1< E O¬ Dx/9$//9 O/9y<  -=
E 2/1y3 <= 61© E|  * -.-I/9>9
² :, /9y -.>vE,61-0 >p= >
m

S .PE

- >1E6
f =E < -0W;/1¥ -0> ./1

611-.=E -.¥I 
i  9 ' :  ­ ^—=  M   M   I˜ S < -.=
Ey/9W63/9G/1¥ < >1< $/9$< -.7
.Ey/9L>E U

∼ F ) = Fp [x]/(f D q¬1-. S Fp +

M 6/1

,<=,E6
¥E

HG>1E(W< =6©E=|E©61[ C¥=-.-.,/9>9
.²=: < /9G-i>

f

[/9H, ¥1-.=6 =3<®/9|<= -.7
.E$HG¬ , /1y61 >=6
P]
.° HLEf

1-.:––: <8X >|/9  GW< =6©E= t/9¥ -.>”61vEfB9 E|


,61hE

(n − 1)/2 m


9/9 F * D,> F]
 -I 2=-.-I/9>9
.²=: [ 2/1L -.>YE61 -*>Y >3<5 >1E

n

3 <= 61© E| Y/9$-i>p61\EfB; E

/9|X>1


,61hf
.9/9 P P >3 O]
.-.,

n−1 S  <[¦f
¥< S 1-.1> -iE m -–B9 E /1E,6
.oE
.-.61/1

=-.-.,/9>9
.²=: [ x/9\ -.>œE,61-y >«= > EC
. /9b/9

n

M B;I C>9
i6/1OP<  -=
E*> <=[/9I61OI 
.6/9[/9

-iy 23 <= 61©E  -i ~



h f
i[ vX>1

1)/2

SyU

h



f

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S(y) = S(x + e) = S(x) + S(e) = HxT + HeT = HeT  6/9

= H i1 e i1 + H i2 e i2 + · · · + H ik e ik = H i1 + H i2 + · · · + H ik , Hx

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i

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&/"  > $   -67 >  -67 >  "!

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C = {000, 110, 101, 011}

S

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 ƒ ‹iŠ M 

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c = (0, 1, 0, 1, 1, 0, 0)         0 1 0 1 HcT =  1  +  1  +  0  =  0  . 0 1 1 0

/1, <=9 E6
\ ² <=[ C\6\:
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c

1-9/9>§

c = (0, 0, 1, 0, 1, 1, 0)

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S q <7?



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9/9 \H| C²[ C<   S `A-.fE 
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r+s+t s+t+u r+s+u r 6/1

r, s, t, u ∈ F2 H

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= 0,

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xc(x)

= x(c0 + c1 x + . . . + cn−1 xn−1 ) = c0 x + c1 x2 + . . . + cn−1 xn = cn−1 + c0 x + c1 x2 + . . . + cn−2 xn−1 ,

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 -

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Fq

‰

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c(x) ∈ C P xc(x) S„³  ­  x2 c(x) x3 c(x),S . q . . E C a(x) = a i  xi }∈ i , P  Hx>1E¨ >1¬W i n a(x)c(x) Hx =>1E¨i [a/9iI(x c(x)) C F [x]/(x − 1) : Cq
 -[< ¬1-. < S M
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` <  = S +

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.E FX >

F [x] M
9/9x[/9I<16~t6><=tE q

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F [x]/(xn − 1)

/1O-i>|E|² 61 EG6xq /9I< S `  -D
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 M 3,<=t
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6
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6

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g ŒC‚T‹ .… ŽŠ‚

 Šƒ ‹ˆ‘¥‹Š

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 -,EfoE D
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 -i/9 - S

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Fq [x] c ∈ C #GŠ®ŽŒ c = fg grau(f ) < n − r Ž‹ˆ‘fŒC   ŠvŽŒ ‰  C n−r €®Œ

n

Fq [x]/(xn − 1)



g(x) = g0 + g1 x + · · · + gr xr

grau(g) = r

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=

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· · · g1

0 · · 0 · · · · · 0 · ·

gr · · ·

· · · ·

! Œ‚… ŽŠ‚.…

 0 0 0 0   · ·  · gr

0 gr · ·

 · 0 0  · 0 0 .  · · · · 0 xn−r−1 g(x)

  \ D,>1o/1e-:e/9

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-.Ef

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xn − 1 = hg + r r, g ∈ Fq [x] M [ 
.” E1<=[ cX>1 S n grau(r) < grau(g) F [x]/(x − 1) r = −hg q q E y . D E M
.E OX>1 S grau(r) < grau(g) r=0 g | (xn − 1)  6/9 W< =61© E=
.< ³  ­  S ` -    M CB9=
|>1E c∈C grau(c) < n q X>1 ,E `

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 1 6  h    E   s  X 1 >  S M S  n c = gqM
.E F *q [x]/(x − 1) xn − 1M  = ghH M jbE X>1     i ¥    = <

 1 > E D 
|  n Fq [x] c = qg + `(x − 1) ` ∈ Fq [x]  Sb³  ­  S jb6
.~ S c+ = (q + `h)g f = q + `h c = f g grau(f ) ≤n−r−1 . D E(f 9/9 |Ht]_-.Ef/9|W-OE$w<7
.=1<  O/1 M X>1x .~|3 <= 61©E  O/9 g06~yE -i>61yEfB; E –:<=[/9x,E S n n − r − 1 F Fq [x]/(x − 1) q [x] 1P E¥w1<7
.=<= b/9 < =6,-.E6
O 6/9,3,6/9,6
., M x .¬W- M M n−r g
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mp D
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 1<=[  }~¥W-

c = fg = jb6
.~

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n−1 X

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Ef /9 LE

j = 0, 1, . . . , n − 1 M vjsX> }~  ¢ S › c ∈ C Hy/1 /1¥3 -[/1 /9 H  0 · ·  · · ·  · · · H =   · · · hk hk−1 ·

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· · 0 · 0 xh(x) · · · · · 0

dim C = n − grau(g) = grau(h) = k.
 -,Ef|› S ¢ S š M D i ∈ C 6
.~ s Y (x − αj ) = M i (x).

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+

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+

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i

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² :[ 2 D ¬1-

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F

q (i) HG= ><®|6;w1E-.¥/9 M < E6
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C0 = {0}, C1 = {1, 2, 4}, C3 = {3, 6, 5},

<= 

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3

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., S ` -OCB9,E|<= M

 ` < 








0S S

 S    0 1



0S S

 S    1 0

   + 

    





0S S

 S  =   1 1

   . 



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6 C[*E²=6=Ef0/90 9/9= OH

xf 9/9  C --. G>1E¨,--. S U

2+1 = 3

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7 7  7






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 …‚ ‘Œ†“‚Š– # n = 2m −1   Š ƒ ‹ˆ‘¥‹Š9! Œ‚… ŽŠ‚

g=

 M (1) 0 · · · 0   0 xM (1) · · · 0 . G=   · · · · · · n−m−1 (1) 0 · · · · x M |ŒC‘fŠT† ‚…  Š  q 6 [/9-iE s| 9/9= ¥¬=6- ¥/11LEE=6|/9x E|? 

S +

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< >16

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 6
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 :c/9 .~o/1 D
 6
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i~ m Fm 1, α, α2 , . . . , α2 −1 -0-.-I D,6
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.=6
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EC
-. M M n = 2m − 1 k = n − m d = 3 X>1$ .~f


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),

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m

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c = (c0 , c1 , . . . , cn−1 ) .L 6/9  C-.-., 3 6/9±63 C[ /9 : C
 S T Hc = 0 E(fW<  61©E  M
,E| OX>1 (c0 , c1 , . . . , cn−1 ) c0 + · · · + cn−1 xn−1 

D E|,6
 

(c0 , c1 , . . . , cn−1 ) ∈ C

i

i

⇐⇒ ⇐⇒ ⇐⇒

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i

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α c M [ 
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.-. ®<[–:;-. (1) c M | c HG]_-.E /9|3,<= OE$w1<=
 1<  O/9 S M (1) /9 :[/9 -

 -16$ > $ 

,

7 7 :7

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n = 23 − 1 = 7



1 1  0 1 G=  0 0 0 0

­ pX>1

0 1 1 0

1 0 1 1

0 1 0 1





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.- §0,-. /9-iGH

m=3  0 0 0 0  , 1 0  0 1

HfpW<  61©E pE| 61 Ef
M (1) (x) = 1 + x + x3 C1 = {1, 2, 4} -I 
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7

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α

9/9 O/9 1LEE=61

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-.=§

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3 <= 61© E| |:-.=43  /9-F-i$,6 C 6
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M

S q /y< E,6
.

W;/1E p>3 -v

H h(x) = (x7 − 1)/(x3 +

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+ ¬E f/1o:,-YX>1o ^ 9/9  f/9 -.- S¯K 6
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/9, 6;: < :[/9

x¬ ­ 
 :x/9P -.-  - /9[ A-.-. 8> .6/9x>EtEf
-.=§O =E < -sŸx/9



9/9  2/1(1PEE 61 S q E$6| ,}~¥6
.-.= - M >13 61hE| FX>1 7
.=: \/9

F

S jb6
i~ M  *,<=,E|,6
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2 39/9,E{ -* -.1-., 6
. /9 2W-

+

m−

HL>E(< E6
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α m 1, α, α2 , . . . , α2 −1

[/9H[$]_® C63 D[/9-i-F>1Ef$Ef
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=6;: H, P/1

m

< =61h3 0/9¥
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= >[ AŸ A/tE
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<= 61h

m

 

2m <  61h L]_  E

1PE|E 61 M 6X>6
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i6
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 6
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.

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n = 2m − 1

EµEf
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1 α 1 α3

α2 α6

Hx>1E{< E6
1-. E|=
 :f/9

c∈C

M (3)

m

α2 −2 m α3(2 −2)

··· ···

F2m

, D
x  D M
E

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c

H0yW< =6©E=$E| 61 Ef<®/9

,

α3

S q E



~x=-.-.C?

M (1) M (3) M (1) M (3)

D*O E6
* 

/1=:;[/9

c∈C (1) (3) Hv>1E´ -i/9 -¥/9e 9/9 c¬1=63-.= M M 1¨X>1v C --.= \/9[ y,--. ~”Hv/97]_²[ C <*/9E63 
.-.-¥X>1 X>6/9 `s-:E $ . =E

 q

S



HcT = 0 Pn−1 Pn−1  i 3i i=0 ci α  = 0 i=0 ci α = 0 3 c(α ) = 0 c(α) = 0  M (1) | c M (3) | c,

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

/9>1
² :[ O/9[ 
.=6
 M
.E AX>1 S

S ¤LC46=E

mŒ iŠ‚T‚T‹"!Œ”Ž Š‹[”ŒC‚T‚.Š–p CEe D,62/9e« 9/9 

/1 /1|3 -

H=  6/9



 
 


  
 


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E A,69? n − k = 2m k = grau(h) = n − grau(g) = n − 2m

p=2

n = 2m −1 +

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α14 α3(14)



.

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K D
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S S

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E§-. M G
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1  1  S H =  SS 

α2b

... .S . .

α2(b+1) S

S

S

S

S S

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α2(b+D−2)

,<=,E|,6
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M >„ D ­  M

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  , 

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S q E

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g = mmc(M (b) , M (b+1) , . . . , M (b+D−2) ) M
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t . =E

grau(M (i) ) ≤ m grau(g) ≤ m(D − 1) dim(C) ≥ S `s-:E 2   E(f D,>1 6
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; > 2



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76 7  7

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q=2





 






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S q 6 [/9-iE

 
    





n = 23

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C0 C1

= {0}, = {1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12},

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|C5 | = 11, grau(M (5) ) = 11. x23 − 1

H

x23 − 1 = (x − 1)M (1) M (5) .

q <[ C>1<[E F *3 <= 61© E|  2E 61=Ef

M (1) M (5)

M (1)



M (5)

°

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d8=6<=E,6
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i

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jb6
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E

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(0 0)G = (0 0 0 0), (1 0)G = (3 4 1 0), (2 0)G = (1 3 2 0), (3 0)G = (4 2 3 0), (4 0)G = (2 1 4 0).

,

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+

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2



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