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q f fy { 9 r fl e v nm wu d q w q d Xq vr wm ? d d ? hm Yt dl q vu nm qj fe f r wy nq Xt V{ wu rq Xj r ? f el d e q
7

vu nm Cj f l f wl q d q d d rq d r o dz dq rw qo fl fu j

=
2

1+q



0

+
q
0

+

1 - q k = 1 + q k , 1 - q k = -1 - q

=

-1-q



+

-1-q

1+q q
i k

-1+q

0

=

+ 0 + 0 =
0

=

Å(k ) = (k + 1).

+
k

n>k>0

(u, p - u), 0 < u ( (1), . . . , (n))

q

=

(i+1)

q

-1-q q

(k+1)

n-1

j =0

Å(k)

0

+

+ + T
q

ti,j q ,

+

1+q

Å(i)

-q

1-q
j

0

(k+1)

, i > 0,

0

+

+

(Å(1), . . . , Å(n))

-1-q

=

=

-1+q k q

0

q

Å(k)

0

(k+1)

= =

q

.

(1)

, 1, . . . , n

f fl y nm i l


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'

? ?? CD

? B ?

ED 9 ( ED B A B

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CB ?

&% B $# A D ? 9
17
k =0

lfk fl w du ~m v z~m z f r{ wo Y n{ rl po ~m rz dl f wj

rz wl z rq f h z f qo { q d

? ?" ??

du r{ 9 r hy po nm de y? nm V{ rq fu ?j q f wo Y n{ h d rq j
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fl hm w r{ rq du

?

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Y Y V{ nq pt

w it 5t nm ?

?

??

h z f fl wu q r po m f rl t "q ? C n Yt dl F dz rq d Y 4{ C e f vr wm } z n t f fl u 5 r{ vo rq Cj f l dz q n po f r i fl fu j n Yt dl 9 } rz dl v wm n t r fu j
) = 2 +
-2

r{ vo rq Cj f l f z n o e m
2

r d d rq 4 nq o ? Xw f f g r f rl o d rz $l n pt w $ rq ~ h z fe d r po v n pt w r ? r l v m hg fe f v pm v Ym Xr q w rq d r y r ? w nm Xy r rw po f l fl fu j ? hg fe d v nm u Y nm u X r r w rq u Å vo q
=
(i+1),j

hm t w o ~m rz dl F d rz l Ö
t

q f wj q w p wm

r f d f~q V{ h o p o n

rq y nm Xy r r

r o wu rz q dl w dz 1t

E

d r g e z q w rq du f dl w? rq fu j n t
?


?? % 10 ? )( ?? ? '? %& ? %? $# " !? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?

k ,s

f dl q rq d

GF (2n ).

(1 - 2k mo d p, -1 + 2k mo d p) = ( (Å(k )), p - (Å(k ))), k = 1, . . . , n. (1 + 2k mo d p, -1 - 2k mo d p) = ( ( (k + 1)), p - ( (k + 1))), k = 0, . . . , n - 1.

l

q f )y wg Xq vr wm r

q f 9 q hm f hm z f u

q rl q rl o w rq u
p.

2 = ( +

c

(k ) = 2k mo d q , k = 0, . . . , 2n - 1.

l

q f ny S wg Xq vr wm

GF (q )

i,j

(Å(1), . . . , Å(n)) GF (q )
-1 2

( (1), . . . , (n))

t

0,j

c

p = 2n + 1

n-1

=

+



(1),j

q

k

Å(i),j

T = (ti,j ).

q

,

+2 =

, i = 0,

q

1 1
s

+ 2,

Å(k ) k=s (k ) q=2

n n

f l w ~m }z rz wl rq du } z v m

?

f l w ~m }z rz wl rq du } dz q w "q
5
3

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o ~ r{ Co Y ~ rw poY r 5 d
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87 G7

d pt Vm hy v nm q d nl fu Gj r o fl fu hm fu j d )
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fe

E

dz?r sz d r

nq dz f y ~m vo rwfj ej w ~m rz

G u e 4! wg rw r t q vr z wm S rq j h e rq Vm e rq q dz q o rq wu ru tvr Ym q f f{ r hr v rz 5q r rq f rl nm fj iodl fl ij q m hm Xq wf qfy

~m rz povr Ymr Xr d d ? lq d rz nm f wj r w hy dew f )q fuq wj r r

f ? Xq d fl d fl fe dl )

fg

i l
!0

i r{ fo Ft z f r{ o r fl hy de ) k r w Xq vr nm
? ? ? ? ? ? ?? ? ?? ?? Å
?

?

Å

Õ

?

)?


0

? ? ?? ?

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t

i,j

=

t

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=

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0
+

,

dl w nq z po fe Vm q z de E rwfl r po dz dq r fl l v hg fe n o d Xq r rw

fu j fyq ddl j hg fe t fy q dl rw )o d q b d f dl q d q r po f

l

dz q ?l dl t p rq wu j d r s q rw po r d r rz wu rq 5 j ? hg
E

A n rq ù po ~m fl fu

y Ym t q fe fe d v

w hy r po n nm rq t po nm u Y

9 dl 9j { ? r r dl nm u X

li X q { hm t w q fe fk u d rq Ylwu qYm fr hg fj w rq u

Å vo q rz l Ö p wm p r o n 1t

E



k ,s

z r q wu f q y ) rwrq pod rv nmt q u d w rq qu Y 5l f dr w rq v wm q w f rq y Aj Vl d r r wz pm w Ym fr fl w o v pm v Ym wr s f 5l fl q d r l v m w nm dl v nm ru pt Vm f dl r u "6 d "nm w
r,

q

GF (q n )

Å(i),j

2n + 1 = p 3 p 0 < k < n 2k = 1 mo d p.



E

Å

? ? nm r fj dl dl w 9 nq t z n t j ) ?g du m r l v Xy r qo ef Xy el Òr r{ vo rq Ym Fj Xy ?r oED CB ? dz ÅA rq
q

(mo d 4) r

dq

q = r 2 (mo d p)

nq f )r de s ru v )m Ó j ?t

dl

t

r

r

p

Vm fl Snm Xy l f d rq

l q rq Cr q j Vmo f d d r

rq

n q 0 < k < n q k = 1 mo d p p n 2n r, q = r 2 mo d p, p

= + -1 GF (q 2n )

2n + 1 = p q n = 1 mo d p q

Yo fl Xy w nm { Xy fe pt Vm f rz hm l t


r vo rq Aj hm t r f o r{ wo Y dz { rq f fy dl v nm ru r d w nqq $ o rl fl d hyw
E

w q E d f dl w fu )j pt f r ? wzr q



wg nq nq Ez vr Xq vr A Sm q nm nq
p

o po r rq vr Ym 5r r u rw wm 9

~m fl "z Ymj vr r po r

?y Er dz wm j t

C d q vr Y ? q d wr nq



o q f m

rXq wu qwe wm de r wupo rd qt ez r y d rq j f du o Xq r z Xq wm
?
p
G7

r

p-1

= (r 2 )

n 3 4, p -1 = r 2 mo d p,

(p-1)/2

= (-1)

(p-1)/2

-1

q q k mo d p k = 0, . . . , n - 1

= -1 mo d p.

,

j0

k=s

-q k mo d p

fy

f {v Y w Y

z n Cj r u dl v nm {

m r r wmy Ym f ?r e ru Vm f r d de r 1
FE

o pt r hy

l v z fm d nm

m i n o ? Á Vm rq d wm

q w d r o

rq d r w

w d po dl 9 f rq rq wm u
p

dl Aq qfl r Áj po Xw r
p

d r e v Vm r hy de

po ~m rz y Ym n rq nq t r

wu rq q Cr vo rq " 4j nm ?z f dr n o d rq fl n

q fj? w j t

rq s v q f

r q l w f y wl rq

~m wu r v fu j

rz d r

f wj "r j m rw ? r t n q f ?y

4z f q

wm f v fm t v q nm o h q w

z w wu Xy d rq f

n nm Xq q j dz q

k = 0, . . . , n - 1

rq d d

o l r 5r r wu w

A vw lf dl fvr de r { q w
-1

? e fe 9 d r hy r l dz

q ohg q r hy de

= +
87

hm ?t rq Cml rz fl fu q fe f r{ "q
5

j

u

GF (q 2n )

j


o

vrfl wm y xr r hy Xw v rwnm rq ej ? Xt V{ wu 9 d bm fl u 5 f r wm f dCmvr q w rq f d f2@ ?dl w



r nq wu vr y r rq ?9 fu n

8

j t


E

fz

d r q r rq ? w

fl n rz vl rz
i

t { pm d F l de

f wl ? fl f

rq fu Yw w q z

Cj Y r{ wo Y z r n o

hl nm e 5{ Y fy fl fu dl
E

{ fl s ~j S vr
p
i

f rl ? f 9l hg Ym r


o dn et e Az r f vo nq de z n o q q f u r Ej l re ?t v Ym r 9 m fl
2n-1 i=0

f f w vr wm
i

4 q E q w rq wu q f wu Xq sr n j n vo fe
E

z vr n d E vo rq rq qj n

voYm 5r rq l j Yo wy

wm pt n nm f rq 4 f{ f wu Xq ?r m rq f v pm v Ym Xr q w rq l )z flfe


w d ?

nhm Yot w wy Xq q d w h de

vrw ?rq dq q t rq v

vw znm pm v

Xy f r rw Yo Ym Xr

fl o y nm sj vr nm ? nq o

f w Å we 9

f f l rq f vr Ym

rw 9z po vrr nm t q 4z m rq v }j r dz hy q y nm w rq r t q nq o ùr

de r hy de r fl ?y n f t r{
i
2n-1 i=0

r nq p o

o r z fl y fe f hm n vrrt fu Ym wj
a
j

fy t wdl Xqv nmvr r q r dl d w
b
i

n Yt dl 9 d v m r fl j r u r 9j rq 4 q 9z q d r v nm Cu r r t n o d "q
5
n-1 j =0

r{ vo rq j f l l wz wm v nm f rl t wu dq Xq r

nm wu q f r Vm d q w r q n v po nm f )u r Y{ r{ wo q v w wm rq nwu q t w"q Xq 5 r r r{ fo t
n-1 j =0

d 9 q r d d rq d f dl w rq fu }j rz dl f wj eq ? v{ nm Xy r o 9 q w rq wu q d 9 w )q { w nm fy rl o w q e z r l dl d q rz l r dl v pm r ) wg rq f g r f rl 9o q d r o v pm v Ym wr nq 4t Xq vr nm r t
? Å
?? % 10 ? )( ?? ? '? %& ? %? $# " !? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?

GF (q ).

-1

{, q , q , . . . ,

q =

f (x) =

aj (

GF (q 2n ), n

fz f{ r

dz

0, . . . , n - 1, f (x)

r t j

GF (q )
i

i = Áq j mo d p j

q

aj

j

bi i = 0,

2

+

q

i

q

+

j

-q

=0

(aj )

bi x

j

-q

) = 0.

q

n-1

i

,

}

2n
-q

?j vr

q z w n ù G fd rwu Xqpo Xy ir r t r r

q r ? Yo wy q de qr~m rz fl

q f w rq d rq fu f l rz z

z

y

q , i = , i = 0, . . . , n - 1 i

l

j

?ru pt vw f nq rl o

Vm nm f o n t

o { " fu r wj t

f r{ d 4q n orl d

wo d Y { o rq j

q f fy h z w nm Xy r vo rq j

Õ


l wz Vm d r v nm )u l z fe f r Xy r vo rq j i Y 5l v{ Vm f fl d v nm u 5 l w v ~m Cz r fl n o n t r fu wj 5q v{ nm f l dz rq j f Y nm fj dl ? q w rq wu d q d r w v nm nm Xy }u r o dz 9 q id il n t vr r Ym u Yr d r rq d j Y f{ j fe? fk fl u
T k=0 =
1+q

z rz l
,

fl fu fj Y nm fj dl 9 q w rq wu q d " w nm Xy r o 9 i )l vr Ym fr d e 5 vr ?m
6

hm t w q f

fl fu 9j Xq dr r vo q " w nm Xy r o z r r t n o d rq 4j ù l

fl fu 4j q f ry r d f fl u w rq fu j rq f wy q v wm d eq Å? v{ nm Xy r o q w rq wu q d 4 w )q { w nm fy rl o w w q r z r fy fl dl f wy rq "j m r l v m ? q w rq f d rq du d rq qj dl d rq d r po ~m rz dl f wj sq Ym fr u w rq u 5
Å
k
Õ

!0

?

)?


0

? ? ?? ?

%? #

? ? ? ? ? ? ?? ? ?? ?? Å
?



q

k

= ( +





1-q

1+q



0

k

-1

k

+

q

+

+

0

)(

=

-1-q

-1+q



-1-q

q

t

k

i,j q

1 + q k = Áq

1+q

+

1 - q k = Áq

i

=
0 k k 0

=

=

+ 0 + 0 =

q =

=

-q

=

+

(i+1),j


k

q

q

i

)=

q

(i+1)

(k+1)

q

Å(k)

-1-q

q

(k+1)

0

k>0

=
n-1 j =0
0

+

(k +1)

+
1+q q

+

Å(k )

+

+

+

ti,j q ,

k

-q

(1)

Å(i),j q 1+q q -q

+

mo d p

Å(i)

Å(k)

1-q

Å(k)

mo d p

,
0 j (k+1)

, i = 0,
-1-q

z rz l

, i > 0,
0

,

+

=

+

n>k>0

-1-q
k

q

=

-1+q

Å(k)

+

0

, k > 0.

q

1-q

0

(k+1)

= =
k

+

q

(1)

-1+q

.
k

fl fu ej m

=


r{ fo ?t z f r{ o r fl hy de nm fj dl x q w rq wu q d w nm Xy r o Gl vr Ym r k r 5 w Xql vr nm
1 + 2k mo d p = Á ( (k + 1)), k = 0, . . . , n - 1.

l

q f )y wg Xq vr wm f
1

f l w ~m }z rz wl rq du } z v m
?

q f rl qo { f r{ wo Y r{ h z f V{ u w rq fu ?j q f Y hm it rl o ~ f rq n vo rq Ój Á r l v m hg fe d r po ~m rz wu rq hg fe d r po ~m rz dl
43

'

? ?? CD

? B ?

ED 9 ( ED B A B

? B ?

' ??

ED CB ?

&% B $# A D ? 9
q

ry d f wj

h z f h dz r y r q d

? ?"

V{ u nq o f wl
??

5 D A ! ' ? D 9 D ? &Å ? &? ? ED CB ? ?? ?? 9 ?? ED ?Å A

w rq r fl }y q rq du

fu 4j q f Y hm z de d rq fu f ry h z f V{ d rq j nq pt

it rl o ~ Y Õ j u w rq u 5 d nm ? d
3

? nq o Xw f f g r f rl o d rz l n pt w A rq h h z fe d r po ~m rz dl f wj $l vr Ym br v n pt w ? r ? r l v m ? hg fe f v pm v Ym Xr dz rq d r y r rw po f l fl fu j ? hg fe d v nm u Y nm u X r r w rq u Å vo
t
0,j

hq $ q

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fj Aj d w Ö p

dl q nm wm p

q f by Xy r r

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rq wu q d C rz dl dz rq 1t
E

?? % 10 ? )( ?? ? '? %& ? %? $# " !? ? ? ? ? ? ? ?? ?? ?? ?? ? ?? ?

p = 2n + 1

GF (2n ).

r d d rq w nm Xy r z d f dl w? rq n


k ,s

(k ) = 2k mo d q , k = 0, . . . , 2n - 1.

l

q f ny S wg Xq vr wm

f l w ~m }z rz wl rq du } dz q w "q
5
3

f l w ~m }z rz wl rq du 9 hm po nm y nm t hm o rf flfk yfl du v ~m r{ }zo fz frl r{o n o r 4t fl q hy w deq 4 d q rq f fu Y 9j hm q it w nqrl ?to ~ v Ym Y qr × r{ fo t
2 2n = 1 mo d p,
n/

0
1 - 2k mo d p = (Å(k )), k = 1, . . . , n - 1.

j0

(Å(1), . . . , Å(n - 1))

n

( (1), . . . , (n))

j
t

i,j

=

t

0,j

(i+1),j

mo d p = 1

= =

(1),j (1),j

+

n T = (ti,j ).

, ,
Å(i),j

,

1

k=s

(k )

q=2

Å(k ) n

n

l

o fu j t Õ



?



Å? !?


9

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17

d dz w rq 4m z de o $ rq r wrr jrz gn vo wu m l ? rz rl o vo Ym u ~m fu et rz rq rv wm m wm r

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?

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yÅ p ? Y l ~m z wz nm m f dl w rq n


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k ,s

vo w wm wr Y d Eq rq d

u fu j t

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