Tiểu luận môn toán học cho khoa học máy tính Lý thuyết mật mã và một số ứng dụng
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ĐẠI HỌC QUỐC GIA TP. HỒ CHÍ MINH
TRƯỜNG ĐẠI HỌC CÔNG NGHỆ THÔNG TIN
KHOA KHOA HỌC MÁY TÍNH
TIẾU LUẬN MÔN HỌC
TOÁN CHO KHOA HỌC MÁY TÍNH:
Lý thuyết mật mã
và một số ứng dụng
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CHƯƠNG 1: TỔNG QUAN
1.1 Đặt vấn đề
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CHƯƠNG 2: CƠ SỞ TOÁN HỌC CỦA LÝ THUYẾT MẬT MÃ
2.1 Số học các số nguyên. Thuật toán Euclide
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2.1.2 Số nguyên tố và bái toán phân tích ra thừa số nguyên tố
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∀D∈›hŸ… ihDi•ŸhDi… hDi‹hŸ ihDi•ŸhDi hDi
m@=6HL@^FJ›
DrŸ@=8_HL@^FJSM(8F›UFH++l(@=
8D∈›5Ÿ@R>6J]<wŸhDi;6DFR'hDi(H]l(6J]Sˆ
Ÿ@=
:hŸi•
CD(`(HL@^FŸ+6J]JF<QµH^H]l(
@=(JhŸi•:hhŸƒµi
9
i
%7<P(SME8`((JhŸiH^@=H_@F˜`(Ÿ
2.2.2 Tính an toàn của một hệ mật mã
78.‘1‘5%()M<ZMJQeF>BJF>UM`(6<4
8P5H(J(UFWF(8@=8CDp)H664<48P`(68P
8O5J)H++S684<48P)=)=`(8_8P8OH^H]
l(D(F%)8P8OS = (P , C , K , E , D ).
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.1
rJ6P'P , C=KH^;6H]CT6'E<Z;6DFR p
P
(.),
p
C
(.) = p
K
(.). P>58;∈P, y∈C và K∈K, p
P
(x), p
C
(y) vµ p
K
(K). CT
@=6;6DFRHbSe<Jz@=;5Se<8O@=>=S)6@=*
[6DFR+HUFS5•L5;6DFR`(<Jz@=;S<8O@=>5
H^SeF@='
h;†>i-_8P8OH^@=<48P)=)=5BF8
;∈P, y∈C+'
h;†>i•'
h;iUFH++l(@=<B;6DFR<Jz@=;@=
(Fc<B(>SM<B<8O@=>‹+6S65+MU<8O
SM)(<BQ8U<Jz‹<Jz=<8O566<BF
5@=H_@P'(F(+H]@eD(FHE>
Định lý 2.2.1DrS = (P , C , K , E , D )@=8_8P8OHUFS††•†%†•†*†5
T6P' P , C , K+DZ6'fr<w(F*H+5@=<48P)=)=BF
=oBF8IS)6*∈K H^c;6DFR<w(F@=.š†*†5=8
;∈P, y∈C+8_S)6F>R*∈K D())Y
*
h;i•>
%T8:
a) DrS @=<48P)=)=*H+58;∈P, y∈%+'
h;†>i•'
h;i)=
J((+B'
%
h>i‰08>∈C . }H+Y)MTk(>YD(+'
%
h>†
;i•'
%
h>i‰0UFH++l(@=+4R8_S+(*D())Y
*
h;i•>QP>5BFZ
H];∈P Q(+
| C | •†‚Y
*
h;i*∈K }| ≤ |K|
Y)B`(H]@e†%†•† K|, )H+
†‚Y
*
h;i*∈*„†•|K|
HUF=>+l(@=SMb+(S+(*
.
≠*
9
D())
P>(HOT8H^8;∈P5>∈C+HX8_S+(*D())Y
*
h;i•>
*eF•†K†=Hs K•‚*
.
5{5*
„%ZH]>∈C =Dr P•‚;
.
5
{5;
„5.≤≤!cMTk(>YD(@L+
!)B@=<48P)=)=5(+ • }H+DF>J(
85.≤≤5'
*
h*
i•'
%
h>iP>65'
*
h*
i5.≤≤5HUF<w(F5=)H+HUF<w
.š†K|.
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.2
<ikE>K(T8^@LB'
*
h*i•.š†K†8*∈*5=
8;∈P5>∈C+HX8_S+(*∈K D())Y
*
h;i•>(4
**L>WF(P'S+(KQ
*
h>iL>WF(P'P, )H+
=(H^'
%
h>i•.š†K†8>∈C .
-sS65*@=S+(F>R8=Y
*
h;i•>5(+:
%
h>†;i•'
*
h*i•.š†K|
!cMTk(>YD(@LH^8;∈P5>∈C :
P><48P)=)=]@eH^T8
2.2.3 Thuật toán xác suất
*68FP)68=(KbF@=FP)6RH]5H+@=8_B
JQ6'u')6JV@FHfF=)=)SBWFpHfFJ(Y)
!:*F5FP)6+2F_4C<4VFL5FP)6@FMSBX
D(F8_DZVFL<‹4;6H]58I<`(FP)6'H^;6
H]8_64;6‹P'^'HfF=)=HfFJ(`(8IFP)6mH^
;6H]JzJ=‹=4FWF58'u')6J)FP)6'@=C<5+
bH^4;6J)8_K(;6H]FP)6@=S68
C<HZ@P'JQJ86>45=HOH^DrjJR'q<B
(<B5HZUF<=)6J)B5SM'<()K(mQ8
H^FP)6XH_'TL'4)6R'PH^h(Dd;uWF(
RHU=>J)8_'fD(FiQP>5c6FP)6RH]5HZ8_
DZ<=)6(Dd;u86FP)6;6DFR5H+@=VFP)68=c
V@FHfF=)(<qDF86J]`(8_HL@^FC
T=)H+5K@=6DZF
%6FP)6;6DFRKH^;E>)6<=)6WF>BH]5T
6<=)6;6H]J8_P'^'V@FD())T8IV@F<=)6
+8_J@K+)sSM
K((6FP)6;6DFR=(@)L@)LFP)6-)Y%(J@)
=@)LFP)6(DY(DFP)6-)Y%(J@)@FMSBXSBWF+
)sSMHZ8V@FHfF=)<RSˆ‹AFP)6(DY(DF>m
SBX8V@F5+bSBX8_M<6)SM+J@K+
)sSMFP)6-)Y%(J@)H^@=U+5BF+)J@K+Q
J@KH+GG@=HX5ABF+)J@KSMQJ@KH++bD(
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.|
8_;6DFR¡=)H+C58_FP)6-)Y%(J@)H^@=U
SM5BF+)J@KSMQJ@KH+GG@=HX5ABF+)J
@K+QJ@KH++bD(8_;6DFR¡=)H+%AFP)6(D
Y(D5BF+SBXJ@K+)sSM5QJ@KH+GGHX5=+
+bSBXM<6)SM+J@K8_;6DFR¡=)H+
K((6FP)6;6DFR=(@)L@)LFP)6-)Y%(J@)
=@)LFP)6(DY(DFP)6-)Y%(J@)@FMSBXSBWF+
)sSMHZ8V@FHfF=)<RSˆ‹AFP)6(DY(DF>m
SBX8V@F5+bSBX8_M<6)SM+J@K+
)sSMFP)6-)Y%(J@)H^@=U+5BF+)J@K+Q
J@KH+GG@=HX5ABF+)J@KSMQJ@KH++bD(
8_;6DFR¡=)H+C58_FP)6-)Y%(J@)H^@=U
SM5BF+)J@KSMQJ@KH+GG@=HX5ABF+)J
@K+QJ@KH++bD(8_;6DFR¡=)H+%AFP)6(D
Y(D5BF+SBXJ@K+)sSM5QJ@KH+GGHX5=+
+bSBXM<6)SM+J@K8_;6DFR¡=)H+
2.3 Độ phức tạp thuật toán
2.3.1 Khái niệm về độ phức tạp tính toán
%)8_FP)6yJ<Se¢hT+HfF=)@=6}J)¢i_
'TL'4)6`(FP)6yH^bF@=8_=8DZ£
y
hiD())8I
DZ5£
y
hi@=DZM5(>DZ'u')6DCR'ZH(8=fHbBJQ
4)6`(8QJ6V@F=)+H_=•(+FP)6y+H_
'TL'K(H(T5BF+8_H(ThiD())8H`@(+
£yhi•hi5J)H+£
y
hi@=H_'TL'4)6Y)K(`(y
2.3.2 Hàm một phía và cửa sập một phía
*68H_'TL'4)6FR')(8_6B'P8HZ
RHU<48PJ)6RHU<)8P=()=M!c=>(>(HO+
V86>4Hr+ZH_4)6a=¤'u'48_E>H335
VFP)6+H_'TL'4)6a£hi•9
5Q(>
VV@F+H_=S)•.00056FP)6H+HOSM
b;Y8@=S5Q+HA?S).0/00'u'4gP>58_
'6'8P8O•L+b;Y8@=+H_<)8P()5BFHb8Of
'8_BJQ4)6+H_'TL'JR@!)H+5'6
=Drj6=8DZ+H_'TL'4)6JR@@=+el(BDTWF(
JHZ;E>6'6'U8P8O=()=M
=8DZDZ>•£h;iH^@==88_'4(h)Yƒ¥(>£F)i5BF
4FP};J(>@=¦‡§54^}>Q8@L;@=JR¦S+§5pHE>
64}¦‡§=¦S+§SM+6H]l(4;68=H^bF8_6
=5(+bbF•L‡@=4H^J)K(H(ThH(
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.
T<PR'i5AS+@=SM4H^J)K(H(TgBQ)
HB(>5Q8=T88_=8DZ=)H+@=SM4H^J)
K(H(TA@=JRS+S75)ƯS+ĐKSoH^bF
8_6HC@=(Q8H^FP)64+J)K(H(Tg
6bFCHZP>UƯĐ=ƯS+Đ5K(HOH(J(8_DZ4jD(F
HE>U6=88_'4(
4j.%)'@=8_DZF>Z5=@=8_'frF>FÔ8)'=8
DZ>
;
8)'h}
ă=)
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Q8;8=(SeF;@)
h>i5@=8_=8+H_'TL'4)6JR@
=8>Êh;iH^@==8r(DP'8_'4(hJ('))J)YƠ(>ÊF)i5BF
4FP};J(>@=ƯĐ54^}>Q8@L;@=JRƯS+Đ5+
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9
8)h}
=)
imH^;Y8@=8_=88_'4(
CHNG 3: H MT M KHểA CễNG KHAI - RSA
3.1Gii thiu m u
3.1.1 S ra i ca mó húa cụng khai
J(HK`(S688P8OS)6MS(@=8_B<_+4R
<)sJ)@]Dr8P8O+F5G@UD'6Jb`(S)(
4)6HLK(+b;Y8KHb8SpHfF`(<)sH+@=D
;FRep`(ê!ÊôY=-:Y@@8(H^JQ<=>=)6D6F78
.|L_]WFZ(=78`(yơth)(Si
3.1.2 Mt s bi toỏn c bn
Bi toỏn phõn tớch s nguyờn (thnh tha s nguyờn t):
%)DZF>C5Q8R6DZF>Z`(+5(>@=Q8L
'E44G`( J)H+'
@=6DZF>Z}s'S6
(F=6
.
k=)6=>+@8PB6<=)6r4F>Z(>r4
^'DZ`(8_DZF>5VQ8=(<BHB(>5+K
S+CUFD)(<=)6r4F>Z=4^'DZ
J)@eF>B8P8O5<=)6=>KH^Drj6V@F@=
DZF>k@F85T6DZF>C+L4`((DZF>Z@=)
H+
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.,
Bài toán RSA (Rivest-Shamir-Adleman) :
%)DZF>C@=4`((DZF>ZS((F58_DZF>
CYD())hY5-hii•.5=8_DZF>‹Q88_DZF>8D())
8
Y
—h8)i
UFShY5-hii•.<)H8)8IDZF> ‚05.55ƒ.„+
HX8_DZ8 ‚05.55ƒ.„D())8
Y
—h8)i
!‡R>JwBF<B(}(DZF>Z`(5T@=<B•'WQDd<B-
hi•h'ƒ.ihWƒ.i5=}H+5)hY5-hii•.DdQ8H^•Y
ƒ.
8)-hi5=)H+
DdQ8H^8•
8)P>5<=)6xy+bWFJ)K(H(
TU<=)6'E4DZF>F>Jw)HB(>(+8_T
8=))WF^@LUFKJw(<=)6
H+@=CHC(FUH_'TL'4)6
Bài toán thặng dư bậc hai :
%)8_DZF>@v@=^'DZ5=8_DZF>( ®
5P'R6DZ(+
SeF®()< O>WF>BH];Y8(+@=s<P(Y)8)(>
SM¯
J)@eF>B8P8O5<=)6=>mKH^;uJK^'@=
DZF>k@F85T@=4`((DZF>Z'=W5•'W(XeJw
J)JK^'=>5BF( ®
5Q(@=s<P(Y)8)S=oS
UFS=>+brH^‡=Q+CHCHUFS(
h'
ƒ.iš9
—.h8)'iP>5J)JK^'=>5<=)6s<P(+b
WFJ)K(H(TU<=)6'E4DZF>-sS65BF
SM<B6'E4=}(DZF>ZQ)HB(>5SM+6
=)H^<=)6s<P(J)K(H(TUFH+`Z
8U8Jw<=)6s<P(=<=)6'E4DZF>@=+
H_S+CHC(F
3.2 Hệ mật mã khóa công khai RSA
3.2.1 Một số bài toán cơ bản
CH3F`(8P8OS)6MS(H^)<p
•h5%5*5:5!ih.i
J)H+@=P'Se<Jz5%@=P'Se<8O5*@=P'6S)6*58IS)6
*38+('f*•h*™5*°°i5*°@=S)6MS(=)@P'8P8O5A
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(.‘
*°°@=S)6<48P=)8O8ISe<Jz; 5FP)6@P'
8O:)(Se8OCT>•:h*°5;i %5=Se8O>FP)6
8O!Dd)(@LSe<Jz;!h*°°5>i•!h*°°5:h*°5;ii•;
b;E>8_8P8OS)6MS(xy5(J8_DZF>
•'W@=4`((DZF>Z@58_DZYD())hY5-hii•.5=4
DZD())Y—.h8)-hii
-Is'*•h*™5*°°i5*°•h5Yi=*°°•Dd@=8_s'S)6`(8_8P8O
xyjb)8_K(8(
P>5DCH3F`(8P8OxyH^H]l(<p(D6h.i5J)H+
•%•€
5J)H+@=8_DZF>k@F85T@=4`((DZ
F>Z‹
*•‚*•h*™5*°°i *°•h5Yi= *°°• 5hY5-hii•.5Y—.h8)-
hii„‹
:=!H^;6H]<p:h*°5;i•;
Y
8)58; 5!h*°°5>i•
>
8)58> %
bT?H]l(J@=^'T5('T8Jw8s'
S)6*•h*°5*°°i5=8; 5(HUF+
!h*°°5:h*°5;ii•;
P>5)Y—.h8)-hii(+bBY•-hi….BF;F>Z
5QcH]@e:F@YJ(+
!h*°°5:h*°5;ii•;
Y
— — ;h8)i•;
BF;SMF>Z5Q)•'W5)s;(B)'=F>ZW5
)s;(B)W=F>Z'5=-hi•h'ƒ.ihWƒ.i5J)(JK^'
(HUF+
‹
‹
5T!h*°°5:h*°5;ii•;‹
4jDr•'W•9/2•922.•|0.9•0•5(+
-hi•h'ƒ.ihWƒ.i•9/2|9220•|00•,00
%Y•/|•1‘ 5=4H^•199.‘.D())Y—.h8)-hii
-_Kcy+bS)6MS(@=*°•h•|0.9•0•5Y•/|•1‘ i
=VS)6<48P*°°••199.‘.-_HZ6k8FZr)y8_M<6);
•29/1|•/5 Dd c S)6 M S( Hb L) < 8P 8O > •;
Y
•
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(90
29/1|•/
/|•1‘
8)|0.9•0••/|20209yPH^>58ODdH^<Jz;
•/|20209
199.‘.
8)|0.9•0••29/1|•/
3.2.2 Thuật toán sinh khóa
bDrjH^8P8OS+(MS(xy5J8IK'L)
J)8Q8_s'S+(38S+(MS(5=S+(<48PL)J(S+(
MS(=S+(<48PY)6<D(F
ƒJ(9DZF>Z@'=WFh'£Wi
ƒ4•'¨Wƒ-hi•h'ƒ.i¨hWƒ.i
ƒ%8_DZYD()).ŠYŠ-hi=@=DZF>Zc(F
-hi
ƒ4D())¨Y—.h8)-hii.ŠŠ-hi
3.2.3 Thực hiện hệ mật mã RSA
Phương pháp mã hóa RSA:
•'¨W'=W@=(DZF>Z@v'E<
%)•%•€
=H]l(
*•‚hh5'5W5(5<i•'¨W‹'5W@=DZF>Z5(<—.h8)-hii„
8IS•h5'5W5(5<i∈*5đ]l(
Y
S
h;i•;
<
8)=
S
h>i•>
(
8)5;5>∈ €
6J]=<H^M<Z5J)S6J]'5W5(H^V<48P
Sử dụng phương pháp RSA:
6D(DZF>Z+6J]@'=W
4•'W=-hi•h'±.ihW±.i
%F8_DZF><h.Š<Š-hii?(h<5-hii•.
46J](•<
ƒ.
8)-hih<wFP)6:F@Y8pJ_i
6J] = < H^M<ZhS+(M_i5J)S6J] '5W5( H^
V<48PhS+(Ji
b8P8Oxy)8_8LJF>U<)8P5)=;E>
6CJQ4)6=8:h(8<BHfF=)@=5Y=;i==8!
h(8<BHfF=)@=5=>i5(A')8IK(8(8_<_
h5Y5iHbL)6S)6MS(*°=S)6<48P*°°8O`(8IK(8
(o+S7<)8PS•'W@=DZF>JR@h=)H+'5Wm'@=
VDZF>ZJR@i‹JR@+l(@='5W'+<bF‡P''E+
C.00VDZ5)H+++C900VDZP''E5(>•.0
900
g
4)66DZY55(>6=8:5!5HUF`>BF@=6
'u'4DZJ6DZF>JR@‹URHU=>J)8R>j78WF(5
S)(@P'JQ86>4HOHU;FRUFCJQ86>4@=8JR+F
WF5(+b(8S)HbDrjS68P8OxymUF
8P8OS6
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(9.
3.2.4 Tính bảo mật của hệ mật mã RSA
_()=`(xyH^BSB(JH_S+<=)6'E4J(
}(DZF>Z•'¨W9DZF>Z<48P@'5WBF(6DZ'5W
S).00VDZP''EQ+Dd+S)900VDZP''Eb'E
48_DZF>a@B6FP)6(R(>c
V86>4HLRm8R=JF78P>'E4DZ
F>=6}(DZF>Z'5Ww88jH4<vO>8P8Oxy@=
HUFS++b4)6qBFJ)WF6JQBSBxy(DZ
F>@
-_8LJF>U<)8PDrjDCH368P8OxyH^;Y8@=(
)=5BFFE`6HUFSC<8IK(8('H_@P'@(
6(8DZ5Y5`(J8Q5m+l(@=6}(DZ'5W`(
h•'Wi5=)+'5W4H^-hi•h'ƒ.ihWƒ.i5=}H+Q8H^Y5C
HZ‡=‹m4QP>8=D(FSHOQ8IK(8('
VF>HZ<48P66J]'5W55oM<ZS)6MS(h5Yi8=M
3.2.5 Chi phí và tốc độ thực hiện của thuật toán RSA
Chi phí: bFP)6xy'f@'Z'46
'u'4C<L)S)658O)658ONF6JQ8O)6=8O
CHC'46'u'4@F²}(8)F@YbH8<))
S)6<48PH^()=QKDZ8mMS(Y?CUFD)
DZ8m<48P!)H+'4K(Hb8O)6V@F?CUF
D)K(8O
Tốc độ:ZH_`(xy@=8_J)VHb8>BF`(xyD)68O
HZ;T5D)8O!:QxyP8C}.00HB.000@f5QP>xySM
H^cHb8O)6SZ@^V@F@8=KcHb8O)6VV
@F?
3.2.6 Một số phương pháp tấn công hệ mã RSA
a) Phương pháp sửdụng φ(n)
DrKRM<BH^6J]-hi*H+;6H]6J]'5WH^
H(U('CJQD(F
•'¨W
-hi•h'±.i¨hWƒ.i
(>W•š'5(H^'CJQ<P(
'
9
±hƒ-hi….i'…•0
'5W4@=(8`('CJQ<P(=>F>RHU'6
H^6J]-hiAS+C;6H](}(DZF>Z`(
b) Thuật toán phân tích ra thừa số p - 1
Thuật toán phân tích ra thừa số p – 1:
P'=k
.(•9
9£)J”•9)k)
!I7C !F>GE
bF@FP)6)*-eF>B8P8O=8_DZTj J(99
(•(
”
8)
/•h(•.5i
1£.ŠŠY
@=}(DZF>Z`(h=Mi
Y@DY
SM;6H]H^}(DZF>Z`(hR<Li
FP)6)@@(J'ƒ.@=8_J)VFP)6HCFWFc
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