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av j Tu i Bn e & j wu 3l 3f Wo U q s U i 3v h ps& U q W rm u q lçy un j iç u u w n g#u ri u q j ri rmri q qq n im i r i w içv @u jl j4u k au nt f Qs o d r q m j xu i v jl Un u q !njl U ru Bi m ç h i ~
C DP
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è éè çå æã å äã â

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s s s s s s ss ss ss ss ss ss pu f i Un q l f g v y U g v y U m u wvn Ui u u á al &
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r

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k h w i çv l j n au l q g i n o
r

B1 , B 2 , . . . , B

q n

A GF (q )

r

f



(fA (B1 ), fA (B2 ), . . . , fA (Br )) l S L4 (U ) gl

v q n

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3

i

n n s n U q j Ui Bi i
R

f Wn v g i n o m p gj q Wi u f j Bq ru u& Wnfq fv h i ri gUn q iq 3 n o mm r U f q p u Wq U Uv q r n q !o r U u q g Ui i j Uu u z q j g u }u s s s s s s m uwq k P w a l& Bn W d U U q r q r Wy hk w i çv l j n au f l ff l
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u = A(A).

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r B1 , B 2 , . . . , B

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D Pr,gl B = (B1 , B2 , . . . , Br ) Gl =

D Pr,fA B = (B1 , B2 , . . . , Br ) FA =

1

A

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f4 T as g u i rm Uq Ul n k i j i w o Bu j x 3u pu f h U g 3inm d p6o m m Un r qv m rpi m n u o ms w|& 3y j au l f &i i i gè U èå q bf nu ée f rm Hd q áWi âäf n i i cä ba wi `ä u çv éY j å au fv
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c

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U q U i n p m jl u n wvv au nWf i r u q m u W Un & q p m W u jn v u o m m Un & Pq j i au o f s wv Tu i v U i o h i v Ti d U P jl as ru &~3 f g j ç u 3qj u n U 3ir q r j qu n d U U q s m d m U q i
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A GF (q ) l S L4 (U ) B
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(' ã ãè è è & é% è è #$ #" ! æ S ã éè åç æá åä ãá âá çè è

A GF (q )4 l S L4 (U )

ri x Uk di y

ris U n i i i n w ri r i f n W

k = 3 log r + log n

q 3y 3 Um ri d 6o u m q d y u ## u m ## o B mPe

bp(C D P

ptime(C D P

Qu i U



U q m # U m k q p rm j vv rm n iy
R R R

j Tu n o q U ip Qsm Ud q d p U

bp(D P

ptime(D P
r,f A r,f

mem(C D P

DP

) = O((r 13 (n - 4k )5 )(log r + log (n - 4k ))2 ), i = 1, 2, . . . , r
r,f r,f

) = O((r 25 n4 )(n - 12 log r - 4 log n)5 § §(log r + log(n - 12 log r - 4 log n))2 ),

r,f A

) = O(log r + log n) + extime + crypttime,

U

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) = O(log r + log (n - 4k )) + extime. ) f 2 B
n 7

1+

n

4

X

C DP

1 n

B

1+
r,f

f (Ai , Bi ) = B6 (A, u, Z , Y )

GF (q )

1 n-12 log r -4 log n

YA = FA (BA ) Zl = Gl (Bl )

BA = B3 (A, A, B , u) Bl = B4 (l, A, B , u)

B

5

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n

rr

. 2
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n q

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ptime(D P

= O(r 25 n4 (n - 12 log r - 4 log n)5 (log r + log (n - 12 log r - 4 log n))2 ),

u Um i jl u n U Hi m jf }u s
0



ri q y Qu j Tu U q y

bp(D P
r,g l r,g l

= O(r 12 n

bp(D) = O(2

bp(D) = (|GF (q )4 | + |S L4 (U )|) § bp(D P

ptime(D) = O(log r + log(n - 4k )) + extime =

|GF (q )4 | = 2

k = 3 log r + log n

4

) = bp(D P

bp(A) = O(r 7 k 2 ),

1+

) = ptime(D P

bp(C) = bp(D), ptime(C) = crypttime.

4k

4k

= O(2

1 n

1+

= O(log r + log (n - 12 log r - 4 log n)) + extime.

|S L4 (U )| =

r 13 (n - 12 log r - 4 log n)5 (log r + log (n - 4k ))2 ) =

r,f A

4k

1 n

) = O((r 13 (n - 4k )5 )(log r + log(n - 4k ))2 ),

1+

r 13 (n - 4k )5 (log r + log(n - 4k ))2 ) =

l S L4 (U )

r,f A

1 n

ptime(A) = O(log r + log k ). |GF (q )4 | § |U | =2 |GF (q )|
r,f A

) = O(log r + log(n - 4k )) + extime.

r 13 (n - 4k )5 (log r + log(n - 4k ))2 ),

)=

4k

§

|U | 2k

DP

r,f

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j j i Un o u m n u

Qs q &i r #m ri

j Tu qq j &3u x

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s

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0
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r,f

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(' ã ãè è è & é% è è #$ #" ! æ S ã éè åç æá åä ãá âá çè è

O(1)

= O((r n )(n - 12 log r - 4 log n)5 (log r + log(n - 12 log r - 4 log n))2 ).

+ O(log r + log k ) + O(k ) = O(log r + log n) + extime + crypttime. = ptime(D) + ptime(C) + ptime(A) + ptime(B 5 ) + ptime(B7 ) =

u rm q n i w u Pv f g i q W v i i & i l ç U 3 i r B l
bp(B6 ) = O(|S L4 (U )| § k + |GF (q )4 | § k 2 ) = O(k 2
4k

bp(C D P

ptime(C D P
r,f 25 4

bp(B5 ) = O(|GF (q )4 | § (r k 2 + r n)) + O(|S L4 (U )| § (r k + r n)),

= O(log r + log (n - 12 log r - 4 log n)) + extime + crypttime+ bp(B7 ) = r § bp(B6 ), bp(B5 ) = O(r 2 ) = max(bp(D), bp(C), bp(A), bp(B5 ), bp(B7 )) =
r,f

)= |GF (q )4 | = 2
4k

§ (k 2 + n)) + O(r 2

4k

C DP

ptime(B5 ) = O(log k + log r ).

ptime(B7 ) = ptime(B6 ),

|S L4 (U )|

1 ), n ptime(B6 ) = O(k ). 2
4k

§ 1+

§

1 , n

§ k+

k ). n

C DP

r,f

v


m Po

v U p m Qu w Qt Qs d Uk q U j j u xu q am r xv m ru r Um r ri m q pi Bi m Q v U k q j ri u çq @y r r m q ä au h f ç r vs Um y i n s ji Wi i &f n iU g i i n o g wv m ru d & l ç U P jg Bfu m n jv i ps U jqU uq U n i f 3i vi çi ( i m av jh 4u rt rm ps q U q nU ii jvav i r rU 4q ç n W o k( xuri
E

äv vwu jl nru (o u rq q r jl s v j Ti 3u U i v u #i rm wu q mo W i U riv U q åm r u q r r U q n q wv Ui Tu u i i n @o v r n U o q äi u y j 4i ç e f j uç ng U4i 3i d i U y v w &u p U m pi B m u pu v Ti y xuU qw 3 3u}y u i pi Tn P w|rm q Wy f d n i g &s
)( '

Un q u n n i h r w q u çl u f dmj uu qn i d 3 W j xu #v U jl ru çiWi Hf wu 6l Um i m 3 u pi n i 6 l m B i Wy i k iwv wu #l u i y wèi j u xu è rv jl i j ru nTu df
9 FH )G

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m u Wy j i o v j ri q 3 ri h q y çv Qu jj Tuau df U ri q q jy yu m s r Um i o r r rm q n i w u i o r q Tu y jl r u & Bn z f g g @i j 4u Pq g U
è éè çå æã å äã â

m = |GF (q )4 |, p = |S L4 (U )|

mem(C D P

2

4k

1+

r,f

1 §2 n

)

n-4k

(|GF (q )4 | + |S L4 (U )|) § 2

1 n - 4k 1 1 1 1+ = 2n 1 + § 1+ = n - 4k n n - 4k 1 1 = 2n 1 + § 1+ . n n - 12 log r - 4 log n

mem(C D P

mem(D P

r,f

)

r,f A

DP

DP

dg

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u Py y d U i i j U n i i @ çh èç &s

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0 $!

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DG ! $#

u rm q n i w wu U q Ui Ufq k y j Wi j fuu 3n u åi Y j uii (v ii Un ij o i n dm Bo i j i xu çv v j au jl dfru & çf Pd l ri q n ä ri Tu iq WiU çfk w jll u u m n j xu çvr m wv u Qu dm w d u r jn m u j @o xu U v j y wuxu jll wv ru wu l j g ug i in n n o 3i äo ri i ri U U qi q U j y xu y j# j Wi u Wo U w q u U jn i u v o u uy xu n wv wui l i ri 3h &s & æå q s Uk jq wu jl au f i äv k U i w Tu
r,f

DP

H $#

r

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h

n g i n äv o ri Wi q Uf y u 6 m n 4i e ( auç Yf j pu u ç h n r jU q ui n ui U i n rm Po 6o q d m n i rj Pf m Un q pi ç m j u Bf U i dv U k w j u U i k y w iu u a m m U y q W p jf rm u r &q rq çh & äã âá c Ui Un Pq 3 ri q
C# 0

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f g i p m jvv lri U q ! i wv p w Uum j Un 3u ( au f u rm q ns i p Bá s
DP
r,f

Um i w|i

r g j pi fy w Hu Um

4m xu u v i j jl

j u ui U k 3u u n U

n U rpi jq m i h Um

Tu v i & n
n

Ti v o W

u rm q U i } pu mv
f

I @ !P & 6 0 0 4 3 E 'T & 6 V6 b3 0 6 ãâ Q å & E & Q B& é3 !8 & Q $ & æå äâ !2 !& !2 & @ãâ Q 4T S3 Q E #" 6 V6 U3 4T S3 RQ é3 $ 0 #" 4 é3 8 #" 76 5 4 3 2 10 )( 0
n rr 2
n 16

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av wv u

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4

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2

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m j n xu n

v iu än i

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r jf vq u jlr (q i

u 6n n i
g

n UmUn #o

qi inu g

(ori rm Uq q m3 u rin B äi i i w ri u i q & i n (o ri U q s
r,f

v k r jq Tu i v y s r Um i Uk n jqi 3u gU i v u jl u 4n Uk q u By ri q ç g
) ã)

n o q & f n W 6o B dme j U au
(j ) l

pi Bm h r q u jfk u äq v jq xuig çv f j W ju B 4u m åe r Tu s U 3 Ww
DP f



(' ã ãè è è & é% è è #$ #" ! æ S ã éè åç æá åä ãá âá çè è

bp(A) = O(r 7 (log r +log n)2 ),

n i u i Un q pi Uk

w n i mq

äu u

u j 3u y g ji u n g i gè i
I

2 0 V6 b3 0 I 76 HD #" !

é3 ' 7& 9F

!8 & & I0 #" ! 1

@ 16 )( & T Q$ éâ UT 2 0 Q BE é3 ! !D 9 #" ! B A Sè á åä â èç å ãâ á
DP O(r 12 n4 )
r,f

j U y

u o i u q Ui 4 u s (n ir n j Wi f u Um h i i

u m i Un g m jl u n y U xi
DP

|GF (q )4 |+|S L4 (U )|

f A GF (q )4 l S L4 (U )

U yU i ri Qu U d q y j xu j v Wi af u jl iru l n yu g f W B e

ptime(D P

= O(r (n + log 2 r + log2 n + log r log n)(log r + log n)) + extime.

bp(B5 )

r,f

ptime(B5 ) = O(log r + log (log r + log n)).

T U äv l ( q ri q U i u Wn v g i n r sUs v n n xo i r q Wn v g y i n j (o u n h ri i q q r 3u m UUn # Uk n q o i n g i i
DP
13 4 r,f

U ii ri

nn i v g i i

i o g j i 3u nj wu lo ri pi U
13 5

n Bo q ri qm v Tiy

q x y m Um Uni U j i Wi f # U qi
fa

r,f

)=

O(r 12 n4 )

O(r n ) + O(r n (log r + log n)2 ),

1 1 (24k +24k § ) = 24k § 1 + n n

ptime(A) = O(log r +log (log r +log n)).

(j ) A

: {0, 1}

B

5

ri o v i o q y u j u n i f xu 4u r m
I

q f W pi m y j u wu jl älau

èå bf ée Hd á â cä ba `ä éY å X

ä i on riåi 3

i n U rq pi #m

ri Uq m ç y Uk q

Qs q p m j q T n i

r 4 ju (Bi U i 3

i n rm doq r g nU iq w uy mr ä Ukjf }uq y
0

n-4k

16r 12 n

{0, 1}

1+

1 . n


Tu U q y j Wi f y j u n !i i v Wn v g i n o h ri q U Wi U jf q u U q i 3 d i # i n f i g g i n o
f (Ai , Bi ) = B6 (Ai , ui , Z , Y ) = =
lS L4 (U )

i i åv n hU q U 4 li uU #j um u u q i f n

ri Uqp ç m r Ul Uv k u j nq äo j q i ä o qU # n i dv n i iW o v ig i&o n i o
d

s w| y i o U di } s @ i i y n i i w u g i n so q fsj iW o o U v i o i h n i U n g &o i nu g j i gi ni o ri n U Uq o ri r U mq Bv Un au y u j Wi f i Uu i i i o i m m o n ç pi m m i U v y
è éè çå æã å äã â

ri n i äg Ti y

l u n Ur U U

A GF (q )4 l S L4 (U )

BA = B3 (A, A, B , u) Bl = B4 (l, A, B , u)

s

|S L4 (U )| § bp(B4 ) ptime(B4 ) |GF (q )4 | + |S L4 (U )| O(r 12 n4 ) 4| |GF (q ) |GF (q )4 | § bp(B3 ) ptime(B3 ) |GF (q )4 | ptime(B3 ) § bp(B3 ) ptime(B4 ) § bp(B4 ) B3 B4

ptime(B5 ) = O(r §(k 2 +n)§(log k +log r ))+O(r §(k +n)§(log k +log r )) =

bp(B3 ) = O(r § (k 2 + n)),

bp(B4 ) = O(r § (k + n)),

= O(r § (n + (log r + log n)2 ) § (log r + log (log r + log n))).

s v j i o U ( r m n Q Qs r q r Qs l d v j Ti xu v jl ruk Uç i f w Tu ç j j u

R

u Wn Un @q P u j n u U q @i r P q v
B

v r gq i n o j y i n i g s i Tii n i i ng io n u
P

i n i g i n 4o r U q y j Wi Hf Um i y j u n i n o u Um i jl u n U i m jf }u
ptime(B3 ) = O(log k + log r ), ptime(B4 ) = O(log k + log r ).

bp(B7 ) = O(s § r 13 n4 (log r + log n)2 ),
QT `Y X WV U T

3u U h U q n n (o d U i jf rm#o qi n i y w

r r m m au f g i rUn f m m Un Wy jf m f q äi h f @l W i i i d o ä u aur y v iaq o

i n o y p rm Un 4u q wu

i = 1, 2, . . . , r

Zl § B2 (A, u, l)

|GF (q )4 | § bp(B3 ) ptime(B3 )



A GF (q )

ptime(B7 ) = O(log r + log n).
7
4

Y

A



§ B1 (A, u, A ) .

r U q Hy m Un au fr Um ri r q q iTu xo duu 3 j 3u my r

q rpi m au u j 3u g Pi Uk j q u n Un dq &i

4 u i ri q 4u ç t j 4u t
I
0

0

|S L4 (U )|

wu çl fg n pswu 4l i f ri P s i r y r j n U d 4 n jf 4u i j d d i Wno }u ri o v

go i oo oo o hUq @ q i u ri fq
R R

v in v & i rin u q WiUn
ã)

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Wn vu qq y h @o # #

Wv Wi rn q v








jv i n do r U q y j Wi (f Um i y j u n çi n o u rm q n i jf w au u f i r q r U q vUi nu o@n i r m i n n i g m v n o äi u rm q n i w äu u j 3u ä i ri q n r U q Py v u i U k r jq u y Wi f
bp(B2 ) = O(log r + log n), log |S L4 (U )| = O(log r + log n), log |GF (q )4 | = O(log r + log n). ptime(B6 ) = O((log r + log n)2 § log(log r + log n)). ptime(B7 ) = O(r § (log r + log n)2 § log (log r + log n)).
2

ri q y u m u Wy j i ço xu dv r q

f g Bi & W v m p m r m j| äy l
O(log |S L4 (U )|) O(log |GF (q )4 |) B6

â4

jv U dv jvjv r i jv r i vs v wu jv s i

ã)

ptime(B7 ) Zl § B2 (A, u, l)

o Uw 6 v r r U çu Tu Wq s H g #i h m Un ( r y mu Un i o@o i r i çv jv aui f i Bo n Wu i q r q l ri n q i u a v Ti y j u

i wq nn nm r Pq

j i 8~ wv i Um i ur w i

m gm wu o y n

i

Un fi xo r l q q i j@ u k n n o åi y q
A


jf au äf Uk pm i U k r jq Tu i v lam 6 3v rr P i n Po d j wu jl au UWf y U Um B i 4u n i Tt as u o g Ui @q n ri Pi Yq i u po iu Win ijf u dv g i} n s i w Tu y U & y u Uk Unq jq wu & çl i @ 3 i n i v r l g i i n n o o r3u # Uk U q n èo v v n ir o n äili

q s i ifo o i i ri i n

g i i 3 pi y y U U ä r u m( uUn çv r pj au u f Um i u jl u n u U im çv j v au i f j i v u o U @qu o i U n Ui i è g ki U n i o v w i g äo j i wu in l èo i pi çvri m U j q au v PfTi

u ri i v r riPq Tu }u ui y U
4

q Wy m r l ni Ul k i j Wi

u Uk jfm }u i oi n j i Pf r

pWn m v Un g i npi Po m h jf Unjq wud s jl wu au èl f i Bv Wi f n o i u 3 r @ n vi ru hri kUq U å i h v r l v r i riTuw Ul k m j o y U 4 U Um i ri q j wu i jl o au 4u f r m
(0

(' ã ãè è è & é% è è #$ #" ! æ S ã éè åç æá åä ãá âá çè è

DP

r,f

ptime(B2 ) bp(B1 )

bp(B2 )

Y

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bp(B1 ) = O((log r + log n)2 ),

Zl § B2 (A, u, l) bp(B2 ) § ptime(B2 ) YA § B1 (A, u, A ) bp(B1 ) § ptime(B1 ) max(bp(B2 )§ptime(B2 ), bp(B1 )§ptime(B1 ))

ptime(B6 ) = max(bp(B2 ) § ptime(B2 ), bp(B1 ) § ptime(B1 ))+

|GF (q )4 | |GF (q ) | + |S L4 (U )|

+ max(O(log |S L4 (U )|), O(log |GF (q )4 |)).

ptime(B2 ) = O(log (log r + log n)), ptime(B1 ) = O(log (log r + log n)), § B1 (A, u, A ) ptime(B1 ) |S L4 (U )|

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