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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1.2 Ý nghĩa
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1.3 Nội dung thực hiện
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o ZF>Z='E4J(}(DZF>Z
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2.1.2 Số nguyên tố và bái toán phân tích ra thừa số nguyên tố
Số nguyên tố
Ðịnh nghĩa:ZF>Z@=DZF>@C.5SM(B)DZ=)
)=.=4+ZF>@C.SM'@=DZF>ZH^@=^'
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Thuật toán tìm các số nguyên tố nhỏ hơn hoặc bằng số n.
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bF@FP)6)*-eF>B8P8O=8_DZTj J(.0
J)6DZA@LmDd@=8_DZF>ZB'jWF6JQJ5(L
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Bài toán phân tích ra thừa sốnguyên tố
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%+8_WF>JQS6HCU8sepHb=>o@R>DZ
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2.1.3 Đồng dư và phương trình đồng dư tuyến tính
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2.2 Xác suất và thuật toán xác suất
2.2.1 Khái niệm xác suất
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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
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p
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h;i•>QP>5BFZ
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2.2.3 Thuật toán xác suất
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JQ6'u')6JV@FHfF=)=)SBWFpHfFJ(Y)
!:*F5FP)6+2F_4C<4VFL5FP)6@FMSBX
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RHU=>J)8_'fD(FiQP>5c6FP)6RH]5HZ8_
DZ<=)6(Dd;u86FP)6;6DFR5H+@=VFP)68=c
2.3.1 Khái niệm về độ phức tạp tính toán
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2.3.2 Hàm một phía và cửa sập một phía
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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):
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'E44G`( J)H+'
@=6DZF>Z}s'S6
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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) :
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CYD())hY5-hii•.5=8_DZF>‹Q88_DZF>8D())
8
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=)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
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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
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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„
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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
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-_8LJF>U<)8PDrjDCH368P8OxyH^;Y8@=(
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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
kg
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kg
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9
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H]H^(• |9099.192p</`(FP)6=;6H]H^6J]
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9@)
9
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bF@FP)6)*-eF>B8P8O=8_DZTj J(9/
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9
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9
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9
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(+
;
9
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9
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rj@eF>BDZJF)(5X(+b;6H]H^<Z7<P
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BFH^¥@=<_DZ`('(>WQp<9`(FP)65X(+
b'E4H^J(}(DZF>Z(>BF¥F>Zc(F5X
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If —.h8)ithen
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:£
While Š‰.h8)i)
0
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9
8)
if
0
—ƒ.h8)ithen
%R8TFP)6hR<Li
Else
4;•h
0
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%R8TFP)6h=M;•W(>;•'i
End if
End While
d) Bẻ khóa dựa trên lặp phép mã
(mXeJw=8@P'8O£h;i•;
Y
8)@=8_'u')6]JP'€
*
h;i•;
Y
8)5BFY
*
h;i•;+6S65x@=SMYRFH^BF<8O
`(x cũng chính là x-sfFP>5<RSˆ8P8Oxy=)m+V
<JzSMYRFH^5H+@=V<Jz;•ƒ.505.8)hQDZ8mY@FM
@FM@=DZ@viK(T8H^JwBF•'W5QDZ6<Jz; €
SMYRFH^@=<wh.…hYƒ.5'ƒ.iih.…hYƒ.5Wƒ.iiQY±.5'ƒ.5Wƒ.
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RFH^F>5K5=)H+'=W5HUFJR@5¤@6<Jz
SMYRFH^+F@=<uSMH6Sb5=)H+S7s'6<
JzSMYRFH^SML)8_F>CH6Sb=)HZc
68P8Oxy
f) Vấn đề số nguyên tố
b<)H8()=)Z8O+(xy5DZF>•'W'H`@
HbSMb‡=B='E4J(}(DZF>ZL56
FP)6'E4}(DZF>ZHO+bWF>BH^6DZF>
+J./0VDZhP''Eib()=5DZF>Z'=Wf'H`@54
jJ.00VDZRHUHsJ(pHE>@=WF>B<=)6@=8B=)HbSb8
J(8_6(+=4;68_DZF>C@=DZF>Z(>
^'DZ¯
Y)H]l(58_DZF>C@=DZF>ZS=oSo(
B).=hpHE>o;u6DZF>Ci}H+DF>J(5@=DZF>ZS
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DZF>Z¼¹955»5h—0h8)ii
Sb8J(8_DZF>C@=DZF>ZY)'C'6'JDd
H(J(SBWF)=)=4;6F>5K(;r@e`(FP)6Jz
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