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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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A#128)•'K-SL17e•,1TQf45674m12?MM,JV7VI?8 hO145677O7JSA#12815oM?,?>0™T10,4 7kS,45672l\SL1JSA#4567ne<[12SL1 VS,?88O137V1WO)<x4567o4*QN7T 10,Q=75?7V>\1MN378SL1JSA#ne<[12SL1 €?VSQP1,\)1x770Q,J712815oM?, X ,Q;,I,\1W7j?A#4567ne<[ƒ5o(-7O7hO123 SV?Q,H,S4„,•,,g1+18O},18O4m12?M1,\1MN) 701k7125;7?8787V1W1,\1>,JS4567Q,J71+18O O7,HQ@2l1281:,,?Š)1;,->,>07„IfO)M[7 1:,,?-P>H8O17O7QR4g12SL17O71…1O8Q18<,J o [...]... trường hợp này hoàn toàn như trong hệ mã RSA dụng cho số có độ lớn 1024 bit gốc Tuy nhiên độ phức tạp tính toán trong các quá trình mã hóa và giải Đối với biến thể của RSA trên vành ma trận, mỗi văn bản có độ dài mã trong trường hợp này là lớn hơn nhiều so với hệ mã RSA gốc Đối 1024 bit Do là một ma trận vuông nên có ít nhất 4 phần tử, do đó mỗi với hệ mã RSA gốc, mã hóa cần phải thực hiện xấp xỉ... thuộc nên thuật toán lũy thừa nhanh Đối với hệ mã RSA trên nhóm đường cong có độ dài không quá 256 bit Việc phân tích trong trường hợp này đơn elliptic, các công thức ở phân 1.2.2 chỉ ra để tính trong công thức mã giản hơn nhiều so với hệ mã RSA gốc hóa cần thực hiện ít nhất xấp xỉ phép toán Đối với hệ mã RSA trên vành thương các số nguyên Gauss, mỗi Các phân tích trên đây chỉ ra rằng, với cùng... hợp này Việc xây dựng không toán Cantor-Zassenhaus [67]…nên biến thể của RSA trên vành thương phải đơn giản và đây có thể là lý do vì sao biến thể của RSA trong các đa thức trong phần 1.3.3 có thể bi thám mã hoàn toàn nếu áp dụng trường hợp này ít được dùng đến các thuật toán này Đối với hệ mã RSA trên nhóm đường cong elliptic, mỗi văn bản là Đối với hệ mã RSA gốc, do mỗi văn bản có độ dài 1024... tích modulus của hệ mã RSA gốc Tuy nhiên, mặc dù các tính toán trong quá trình mã hóa và giải mã trong trường hợp này được thực [1] D Boneh and G Durfee (1999), “Cryptanalysis of RSA with private DANH MỤC CÁC CÔNG TRÌNH Đà CÔNG BỐ key d less than N0.292”, Proc of Crypto’99, LNCS, vol 1666, IACR, [1 Long T.D., Thu T.D., Thuc N.D., “A General Model for RSA Springer-Verlag, 1999 [2] El-Kassar A.N., R... trong vành thương , thực tế sẽ được tính trên và sau đó lấy modulo toàn của hệ mã RSA với các biến thể của nó dưới khía cạnh thám mã theo (bằng cách thực hiện phép chia cho ) Nhưng do phép chia Euclid bằng cách phân tích modulus Để đơn giản cho việc phân tích, chúng tôi trên thực hiện không duy nhất nên việc giải mã tính có thể cho ra kết giả sử rằng độ dài của mỗi văn bản là 1024 bit và thuật... chỉ ra rằng, với cùng một modulus , việc thiết văn bản có độ lớn 1024 bit nên và đều có độ lớn 512 bit Do đó độ lớn lập hệ mã RSA gốc là đơn giản và an toàn hơn các biến thể của nó Điều của không thể vượt quá 1025 bit Do nên không vượt quá 1025 bit này giải thích phần nào hệ mã RSA gốc được dùng rộng rãi hơn các biến Nếu thì không vượt quá 1025 bit, vì vậy không vượt quá 256 bit Việc thể... này cho thấy để xây dựng hệ mã RSA trên vành modulus là thuật toán vét cạn thương , trước hết cần xây dựng một hệ đầy đủ và duy nhất các phần dư Trước hết, do có rất nhiều thuật toán hiệu quả phân tích một đa thức có thể có trong phép chia một phần tử trong cho , chính là không gian cho trước như thuật toán Berlekamp [49], thuật toán Ben-Or [50], thuật các văn bản cần mã hóa trong trường hợp này... 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(

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