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7/25/2019 Chng 2 - L Thuyt S
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CHNG 2
GII THIU V L THUYT S
Hu ht cc thut ton mt m kha cng khai c xy dng da trn cc khi nim
ca !" thuyt#$% &rong ch'ng n(y #) tr*nh +(y ng,n g-n nh.ng kin
th/c c' +0n 12 !" thuyt #$3 n !( cng c4 h.u
5ch 6 hi6u #u m7t thut ton mt m n(o %
2.1 Cc s nguyn ! "# cc s nguyn ! c$ng n%&u
2.1.1 Cc '(c s
8i r9ng3 b:m7t #$ khc ;< !( =c #$ ca anu a> mb3 1=i m7t gi
tr? mn(o 3 @ y a3 b3 m!( cc
#$ nguyn% 8h 1y3 b!( =c #$ ca a3 nu nh chia acho bkhng cAn !Bi
#$ d% C6 k5 hiu b!( =c #$
ca athDng #E d4ng cch 1it ba%
F cc Guan h #au
o 8u aI3 th* a > I%
o 8u ba 1( ab3 th* a> b%
o J$ b+Kt kL khc ; !( =c #$ ca ;%
o 8u bg1( bh3 th* b:mgM nh< $i 1=i +Kt k* #$ nguyn m3 n%
C6 ch/ng minh khNng ?nh cu$i cOng3 cn chP " nh #au
nu bgth*gc dBngg> b QgI3 $i 1=i #$ nguyngI n(o 3
nu bhth* hc dBng h> bQ hI3 $i 1=i gi tr? nguyn hIn(o 3R* 1y
mgM nh> mbgIM nbhI> b Q :mgIM nhI X3g> IU3 h> VT3 m>
T3 n> S
XIU 1( XVT% Yu cu ch/ng minh r9ng X:T Q IU M S Q VT X:T Q S M S
Q Z X Q IISQ IT%
8uP!( k5 hiu t_ h_ tKt c0 cc #$ nguyn t$3 th* $i 1=i m7t #$
nguyn d'ng +Kt k* c 1it
duy nhKt d=i dBng
I
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7/25/2019 Chng 2 - L Thuyt S
2/20
3=P
appa @ y tKt c0 cc ap;
&rong cng th/c n(y3 +i6u th/c @ 1 _h0i #au dKu +9ng3 k" hiu
t5ch tho tKt c0 kh0 neng ca cc #$
nguyn t$p% C$i 1=i mfi gi tr? c4 th6 ca ath* gi tr? !=n nhKt ca
ch^ #$ ap#) +9ng ;%
V) *+2.- TV;; > SUQ TSQ S%
Fc gi tr? ca m7t #$ nguyn d'ng +Kt kL c th6 !it k d=i m7t dBng
'n gi0n ca tKt c0 cc ch^
#$ khc khng tho cng th/c @ trn%
V) *+ 2. J$ nguyn IS c th6 tr*nh +(y nh aS> S3 aT> I%
J$ nguyn IW c th6 tr*nh +(y nh aS> I3 aT> S%
jh_ nhn hai #$ nguyn t'ng 'ng 1=i _h_ c7ng cc gi tr? cc ch^ #$
_hO h_
k> mQ nkp> mpM np3 $i 1=i tKt c0 ccp%
V) *+ 2./ k> IS Q IW > SIV3
kS> S M I > T3 kT> I M S > T3
SIV > STQ TT
l 2 \7t #$ nguyn d'ng +Kt k* dBng pkch^ c th6 chia ht cho m7t #$
nguyn3 khi #$ +? chia c
+c ca #$ nguyn t$p1=i ch^ #$ khng 1t h'n k3 ngha !( #$pj1=ijk%
8h 1y
abapbp3 $i 1=i tKt c0p%
V) *+ 2.0a> IS 3 b> TV 3 ISTV3 IS > SSQ T3 TV > SSQ
TS
aS> S > bS3
aT> I S > bT%
2.1., Cc s nguyn ! c$ng n%&u
FhPng ta #) #E d4ng k" hiu gcd: a3 b< 6 ch^ =c #$ chung !=n
nhKt :Fp8< ca #$ a1( b% 8i r9ng3
m7t #$ nguyn d'ng c!( Fp8 ca a1( b3 nu
o c!( =c #$ ca a1( b%
o q=c #$ +Kt kL ca a1( b2u !( =c #$ ca c%
F th6 ?nh ngha t'ng 'ng nh #au
gcd:a3 b< > max k3 khi ka1( kbs%
l@i 1*3 Ai hi r9ng Fp8 !( m7t #$ d'ng3 chPng ta c gcd: a3 b<
> gcd:a3 b< > gcd:a3 b< >
gcd:a3 b gcd:vav3 vbv gcd:V;3 SU < > IS%
l@i 1* tKt c0 cc #$ nguyn khc khng 2u !( =c #$ ca #$ ;3 chPng ta
!un !un c gcd:a3 ; < > vav%
w d(ng xc ?nh c Fp8 ca hai #$ nguyn d'ng3 nu cc #$ n(y c tr*nh
+(y d=i dBng
t5ch ca cc th`a #$ nguyn t$%
V) *+ 2.
T;; > SSQ TIQ S3
IW > SIQ TS%
gcd:IW3 T;;< > SITI;> V%
S
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7/25/2019 Chng 2 - L Thuyt S
3/20
&rong trDng h_ chung
k> gcd:a3 b< kp> min :ap3 bp II n > X II > I Q X
M U r> U%
8u a!( m7t #$ nguyn3 cAn n!( m7t #$ nguyn d'ng3 th* amod n3 c xc
?nh nh _hn d ca
_h_ chia acho n% 8h 1y3 $i 1=i m7t #$ nguyn a+Kt kL c th6
1it
a> a nQ nM :amod n U II mod X > T%
2.2 L9 !%uy:! ";
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7/25/2019 Chng 2 - L Thuyt S
4/20
Chng minh &` gi0 thuyt ta c III bnta += 1( SSS bnta += 3 #uy
ra
ntnttbtbbbaa
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7/25/2019 Chng 2 - L Thuyt S
5/20
2.- H#E Fu75
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7/25/2019 Chng 2 - L Thuyt S
6/20
Hy xt tKt c0 cc kh0 neng ca t5ch axi 3 1=i
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7/25/2019 Chng 2 - L Thuyt S
7/20
rbqa += 3 br
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7/25/2019 Chng 2 - L Thuyt S
8/20
3IIS kkkk aqaa += ;< ak< akI
;I += kkk qaa
wa 1(o ?nh !" S%I3 nhn c gcd:a;3 aI< > gcd:aI3 aS< >
b > gcd:ak3 ;< > ak%
V) *+ 2.10 AT%uJ! !6n Fuc7K*B&*m gcd:WIU3 IWX U Q IWX M
VV
IWX > S Q VV M
VV > I Q M II
> Q II M ;
Ry gcd:WIU3 IWX< > II%
&hut ton c th6 1it nh #au
R(o Hai #$ nguyn d'ng a1( b1=i a] b
}a Fp8 ca a1( b:I< hi! b ; do
ramod b3 ab3 br
:S< }turn :a?n% ng%@& 2., A%On !P ng%?c%
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7/25/2019 Chng 2 - L Thuyt S
9/20
&a d d(ng rPt ra dy #au
IIS = kkkk aqaa m( SSTI = kkkk aqaa nn #uy ra
VI%
2./ GKQK 8%'ng !5n% I3 th* !un tzn tBi _hn tE ngh?ch
0o aI% 8n t` _h'ng tr*nh nbax mod 3 #uy ra %modI nbax
V) *+ 2.23D{i0i _h'ng tr*nh I;modTX x %
FhPng ta thKy TI;modX I =
% 8n nghim ca _h'ng tr*nhx> T Q T mod I; > Z%
Z
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7/25/2019 Chng 2 - L Thuyt S
10/20
nh l 2.4 Fho n]I3 6 _h'ng tr*nh nbax mod c nghim th* i2u kin cn
1( !( gcd:a3
n b3 :S%S S3 1* S khng !( =c ca nn _h'ng tr*nh trn 1 nghim3 nu
khng da 1(o ?nh
!" cng d d(ng thKy r9ng SxM I;(> cng 1 nghim3 1* 1 tri !( #$
chNn3 cAn 1 _h0i !( #$ !% Ci2u
n(y !( khng th6%
S% {i0i _h'ng tr*nh TVmodIWV x % R* gcd:V3 TV
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7/25/2019 Chng 2 - L Thuyt S
11/20
8u ta ch/ng minh h trn t'ng 'ng 1=i m7t _h'ng tr*nh #au
nxx mod;
@ y ==
k
iiiii nNNcx
I
I; ?n% ng%@& 2.- AWJcB
Fho gcd:a3 n< > I% lc amod n!( #$ nguyn d'ng nh nhKt tha
mn _h'ng tr*nh zng d
th/c
na modI 3
|5 hiu +c !( rdn:a S SSmod > U STmod > T SUmod > I 1y
rd:S< > U%
>?n% ng%@& 2. ACXn nguyn !%yB
8u nh +c ca amodu!o n+9ng gi tr? ca h(m u!r ca n% &h* ag-i
!( cen nguyn thy ca n%
V) *+ 2.2- J$ S !( cen nguyn thy ca % R* rd:S< > =
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7/25/2019 Chng 2 - L Thuyt S
12/20
>?n% ng%@& 2./ AT%Zng *' [Jc %&KB
Fhon!( #$ nguyn 1( I>n % J$ t` nhm QnZ g-i !( th~ng d +c hai
tho modu!o n3 nu trong
nhm QnZ tzn tBi #$x3 tha mn i2u kin nax modS 3 trDng h_ ngc !Bi
g-i !( khng th~ng d
+c hai% &_ h_ cc #$ th~ng d +c hai tho modu!o nk" hiu -Rn3
t_ h_ khng th~ng d +c hai k"
hiu !( -NRn%V) *+ 2.2D&5nh -RII!( t_ tKt th~ng d +c hai tho
modu!o II%
FhPng ta c { } { }Z33U3T3IIImodI;3Z3W3X3V33U3T3S3I SSSSSSSSSSII
==-R 3 trong 15
d43 chPng ta t5nh -RII+9ng cch !a ch-n cc _hn tE ca nhm QIIZ
%
C7c gi0 c th6 ki6m tra !Bi r9ng { } { }Z33U3T3IIImod3U3T3S3I
SSSSSII ==-R % Ci2u n(y #)t*m hi6u Gua ?nh !" #au
nh l 2.Fhop!( #$ nguyn t$% |hi nh.ng khNng ?nh #au y !( Png%
I% { }Sy
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7/25/2019 Chng 2 - L Thuyt S
13/20
Chng minh C$i 1=i p-Rx th* tzn tBi #$QpZ( #ao cho
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7/25/2019 Chng 2 - L Thuyt S
14/20
U%
=
UmodTI
UmodII
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7/25/2019 Chng 2 - L Thuyt S
15/20
k
kp
a
p
a
p
a
n
a
=
%%%
SI
SI
3 trong tKt c0 cc k" hiu +n 1 _h0i !( k" hiu pgndr%
"#c tnh cht c$a % hi&u (ac)*i
Fho n!( cc #$ t nhin ! 1( a3 bZ% |5 hiu aco+i c cc t5nh chKt
#au
I% 8u n !( #$ nguyn t$ th* k" hiu aco+i !( k" hiu pgndr
S% { }I3I3;
n
a
T% ( ) I3gcd; =
nakhin
a
U%
n
b
n
a
n
ab
%
n
b
m
a
mn
a3 i2u n(y dn t=i
S
n
a!( ; ho~c I%
V% 8u nba mod 3 th*
=
n
b
n
a
X% II
=
n
W%
=
UmodTI
UmodII
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7/25/2019 Chng 2 - L Thuyt S
16/20
M &5nh chKt th/ hai khng th6 m@ r7ng g,n 1=i zng d th/c u!r
pap
a pmod
Sy
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7/25/2019 Chng 2 - L Thuyt S
17/20
1=i
{ }
=
;
II
qZx3
q
3xC %
}[ r(ng
{ } =
=
;;
II
qZx
qq
C %
8u nh ;3 3 th* II = 3xs nhn cc gi tr? trong t_ { }IqZ 3 cho
nn
=
=
qqq
sC
qZs
3
II3
l@i 1* ;;
=
p
3 cAn trong { };qZ #$ !ng _hn tE !( +*nh _h'ng 1( #$ !ng _hn tE
khng !(
ch5nh _h'ng !( nh nhau% &` y
{ }q0q0C
qq Z3
3
Z3
33 ==
;
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7/25/2019 Chng 2 - L Thuyt S
18/20
l@i 1* tho tiu chun u!r3 pa p modISy
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7/25/2019 Chng 2 - L Thuyt S
19/20
2.11.2T)n% cXn [Jc %&K !%6 E6*u76 7# %a8 s
FhPng ta +it r9ng pqn = 3p1( q!( #$ nguyn t$3 nhm QnZ zng cKu
1=i khng gianQQ x qp ZZ %
l@i 1* zng cKu 0m +0o t5nh chKt !nh #$ h-c nn n(x modS 3 tha mn
khi 1( ch^ khi +i6u th/c
trn Png 1=i modu!op1( modu!o q% Ci2u n(y c ngha nu tri6n khai c
nra th`a #$ th* cen +c hai
tho modu!o n c th6 t5nh c tho thut ton #au&hut ton S%I
Cu 1(o S #$ nguyn t$p3 qtha mn i2u kin n>pq3 #$ nguyn p-R(
%
Cu ra Fen +c hai ca(tho modu!o n
xp
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7/25/2019 Chng 2 - L Thuyt S
20/20
S I I;IWSQ ZXSV > I;I ; I;S> II;S; I II;SSQ ZXSV >
XVI
Ry ZXSVTTTmod IIUIT > XVI
