Các tính chất đại số của hệ mã RSA và khả năng mở rộng - NCS. Trần Đình Long

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A04NM(:(#&I45WI-'T4+"4& U "J/!23&Y/.-'Q:" (:,:,aNe4ABC4D6GOP0LQ04G4.a0-8X6LQ04lZ8_U1 1! @?,)•1€1)eI,J1Q1+ @1•?S + 0IQ5>V?70>?,-,‡MS>V?2, G! cUIHM7O7)•1€ cUIH4567SV?1 # 4567,H,SIl7O71+128 (:,:(5cS04L/U8S84ABCNCTUVG JSA#7-7O7JSA#12Q15o7O74?1k7Q Q15o7O7?128)•'(4g1L7Qgo4*12 (:(aNe845GOP00p604DB (:(:,aNe ,H€MSL11N)>O72‘-124V4nO74fSL1)•)18O?, 0,?878Q;,)•)18O-MSL1€?VS-1k7M)•)18O7V1+  >\16)T10,P>+,J1]8M,e,H€M1N)78>O72‘M 1N)7O7QUIH7•SV?1]8>,WA#T10,45?2?SL14,g >,J?878J1k7nH2?125? A04NM(:R#&I-^ a. A;0 I ! /;" b. A;0 ! !& ''" c. _)0/O` F4/:" J$"J/!4 &U %2S" c;,7O7,H1,\15128SJ4g12-7T1?7V1Wne<[ 12JSA#5? ABCGOP00p604DB04q0 1! " X+ X0I\5M>V?70>?,Q,‡+MS>V?2, G! " XM>0,?7O7QUIH7•SV? XL1QUIH4567SV?1 #" X4567,H,SIl7O71+ (:(:(5cS04L/U8S84ABCNCTUVG JSA#7Q7O7I,\1W7j?V128)•'(4g1L7Qg o4*12 (:Rhqd1r0-TUV0G4W8X6 (:R:, Q04s^O-B60 US'^_w}]2S?41,\1MN)4567>\1hH1N)7O71[4*7R1z Q87+VMSL1Q7V)•1€-1284VMSL11\ (.''-’’M,S]1]1?MS;,1,\1MN)45674“7R,‡?Q}]2S?Q;, Q7O7S?12NQ0>+715;7•P1VS1Š17O7>\1hH7• 1,\1QgQ}]2S? t04_u(:,r’’M,S]1]1?MsX24&3C5R5 4?1D ''25a5S35a5*/QR/O` F L1)•1€7V1WQ,\1<5;,<9Q;, ˆ•17O7)•)7,\-Q?,1[4*7R7j?7j?nO74fIƒ,Q c4567S01Hh??,4fMi? t04_u(:(r’’M,S]1]1?Ms4?&% t04_u(:Rr’’M,S]1s_)0/O` F 4?/]" 5QN-S‘,)•1€7j?Q7V1W4*R1Q;,SL1S?12N >+715;7<98m7 t04_u(:)r’’M,S]1]1?Ms241O(" V1W2??)•1€>Hf7128M (:R:(hqd1r0-TUV0G4W8451r6GOP0LQ04s^O-B60 (:R:(:,hqd1r0-0p604DB8acv ,H€M?,1)eI,J1QY+,J V1W>,WS12?45672l)•)e->+,J”•4f‰?Iƒ, MSL1)•)18O?,0,12 }e,:1?4f‰? XOn9  1284V- Q XOn9 1284V- Q V1W1R2lQ8184567nO74f A04NM(:f4/;" A04NM(:'!n9)/OF  4?/:" }e,:7T1?P>+,JIƒ,Q15okM1N)7O7)•1€>H f7128Q->,4VQPM7O7VSe 8,2?->+,J T10,1,\1MN)J1k715o1[5J1k7128JSA# 7125? A04NM(:w \\3+4&U /913%2S " 4[uR:,J4gKK1N12?MJhH7j?J4g(K1284V---Q (:R:(:(aNe4ABC 1! 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\'-U %&Y;0&" }e,:n•1JSA#Q;,7O7>+,J1015:QQ;,4,g>,J7e IlT10,4914567>\1hH? A04NMR:(%4?\'-c +" R:,:R4I0JKGLM`4.a0-`4S`GY08F0-Tg0-1Q046U84UMZ }l1[7,JS7T10,N1R2l>,S131–MJ1R 7045672R17?8-1k7MQ=MSL1Q]7182ŠR17j?<,g MS7T10,‰2l1R70A#Il<?,7,gQ=7„1R14f \1SQ8SL14,g>,J84V \4,g>,J7eIl45671?Il-7T10,N4567>\1 hH\13MSL1Q]7182ŠR17j?< R:(sQUG5S0j9GOP004DB R:(:,zGLY0NM e'4' --f-0ghib+&-\\4' -5-'4\ 4567 h?1eS?1z>,,?81k712?84Z,>V?t,~~,]X]MMS?2?4:,V2R1 ,g70123>H8O17O7)5o)O),H,I,18Ot12SL1 125:‡978125;7c,J7O)<x7O7)5o)O),H,I,18O t12VSe7O7S?12N7„2R1)k719)<8SR1,g1+18O R:(:(sQUG5S0j9GOP004DB ,H€5M1-MVSe7O7S?12NQ07R)( >0I,\J12},18Ot12MI,18O e'V”'3%&1”rKws ,JS17j?I,18O\1*19,P4567nO74f<R11]8 S8<M8Q;,MIN77j?S?12N128VS,H€--S?12N4oQfP >+,JM R:(:R{G.v0-Q;,Nn•12lQ,J7,H,I,18OrKwsP4o,Ho2R1 ,g>,MS?12e7•8-7T10,<[?Q8<97•8’82<?7j?4W MS4o,HV?Q,J7,H,I,18OrKws R:(:)4K54D6|5O1608X6B6GOI0 c,J745?S?12NQg<97•8’82<?I?8*S13SSL1S?12N QSL1S?12N<97•8’82<?t?878tT10,1VS1Š1M9,5 ??1k71]8SL1I,\—  4567@,M4?1k74m71257j?S?12N \>07V,JS128- 1?PSƒ2L1]8SL1,JS7j?4W4567125:Sƒ2L 5QN M07V(,JSr1288m7sV7O7125:6)? ? X?13S7O7Q]71822,kQ;,,O12f2, XY,4VQ IY+,J-7V(125:6)  r,s28125:6)-?13SSL1)•1€7j?Y,4V1* 19,)•1€SQ- r,,sY,4V?13S(Q]71827oƒ7j?MQO7S?12NtQ P4567nO74fIƒ,Q R:(:f4ZIGG5S0N.6TQUG5S0j9GOP0LMGO.]0-8acv y1z?7j?S?12N<97•8’82<?7V1W1+45675? A04NMR:'X214&+Q: !V a. %" b. %" },18OrwwsI?4•7V1W45?Qg<94o,Ho5? rK/s t8S?12NM1+4567Q1+1]8SJ4g12-1?4*R17O71 )•15ok128(S?12Nƒ?,Q\7j?rK/s4W7VSL1J()5o 1231]81SS‘,)5o123MI,18Ot12r8m712125:Sƒ 2L7j?sJ4g?4e>“4fIN77j?7V1W1+45671]87O7 IN77j?7O7)•1€7j? A04NMR:w#&I4 -Ca'! 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Thông Tin và Truyền thông 2012, pp 1 9-2 6 [3 ] IEEE Transactions on Information Theory 36, pp.55 3-5 58 [4] N Demytko (1993), “A new elliptic curve based analogue of RSA , Thuc D Nguyen, Than Duc Nguyen, Long D Tran, “Attacks on low private exponent RSA: an experimental study”, 2013 13 th International Conference on Computational Science and Its Applications (ICCSA2013), pp.16 2-1 65, IEEE, 2013 [4 ] Long D.T.,... based cryptosystem analogue of RSA , 2013 International Conference on IT Convergence and Security, IEEE Ebook, pp 37 7-3 80 [5 Long T.D., Thu T.D, Thuc D.N., “On the heuristic guess of 2- ] dimension lattice attack on low private exponent RSA, Tạp chí Khoa học-Khoa học Tự nhiên & Công nghệ, Đại học Sư phạm Tp Hồ Chí Minh, số 2(67, 2015, pp 10 1-1 08 TÀI LIỆU THAM KHẢO CHÍNH Cairo, Egypt [3] M... Wiener (1990),” Cryptanalysis of short RSA secret exponents”, EUROCRYPT’93, LNCS 765, pp 4 0-4 9 [5] R L Rivest, A Shamir, and L M Adleman (1978), “A Method for Obtaining Digital Signatures and Public Key Cryptosystems”, Communications of the ACM 21, no 2, 12 0-1 26 [6] Varadharajan V and Odoni R (1985), “Extension of RSA cryptosystems to matrix rings”, Cryptologia, 9:2, 14 0-1 53, 1985 . X'.K a( b 8t - t - 7t - ”},18Ot12VS r (-) s - {œ' (- AC5D -% 7D ADAA-9D6n 6- ))'^X(‚ aK b 7t ] - ?t7. '&k776n>3))'‚(X'‚ /- uuu-(.'K aw b 8t - t - ?<7t - ”}]2S?2,I?]< 72)181]S??M8]8~A# - 6n>k - ' 47'M -  'kA7' - . t}8]?<t2~]]r'^^^s-”2)1??M,8~A#v,1)2,Q?1] >]<M]1? .(^(• - i-'"'M 7- 5'mvv3h7 - Q8M'‚‚ - A- #)2,]2Xc]2M? - '^^^ [2] uMXY??2 - A?1?2?<Dv?<r(