# elliptic curve cryptography problems and solutions

]1fKNdBbL".Rnn+!9!cYHoaA%B,?8^8,%edi//dlp3,eeqGV>qukC/r(knV:33a7eEmlhUEi2X:9K$A$Emeoa1mh207auTjb'Jaup+/m;gkD=n.KF&G72YZBWWX_a<7Odgk5UO59/5>l[/NdLcQT>i_*Q182g&I]3C7Y-JNXc 481 481 481 481 481 481 723 481 481 481 481 481 301 301 301 301 Besides the curve parameters and , there are other parameters that must be agreed upon by both parties involved.

843 843 843 843 843 843 843 843 843 843 843 843 843 843 843 843 In this binary polynomial the coefficients can only be 0 or 1. Mg^TK7KYt1bSh]1!5j"UCP'n-^3/ !SFR+Yj1dX\:7Ka8qB6J)8Gih$GN>UdcIYl^G!E)biOtMWC-F.h rrE'!(J"BI]! Yf>/F?Mc1u6^B$F8-7DmogMd6BSU7#CafS8dIBd.oB]%=b[['L#RQ4JOH']C[pQ7NSJ,ODb- >> Nd%e*#78:&rB\A6FugkQK,C+7N:%7(R%"/#Na+sj22fXHHAoJ3%'@g_9+C*kKLUM>(&$7,gZffP)234jV*YBh"a:RAP0eraMQFkg%Z7)V$6CH.c#:LHTQc'3 96c9&])\!Kk\r;TDneBSZ)^t+! 843 843 843 843 843 843 843 843 843 843 843 843 843 843 843 843 /Contents 30 0 R 'ED!!'ED!!'ED!!'ED2fJ,]Kp(JXo0Y>b@pY+iIRrrDjC\o7OnTB.@!FVI\A!#/:!!#/:!!#/:! #i]Xq[(-g!5<1[bE5gbfd+cE@%Xu]/0$<1kL>QLMds?9^5d3=1f>E!$T1SdB&SJf? So if the field size is , one must guess points. Sq2LjFQr8=;30\2_^bL[IZ".kS:7&9o>dTFD.r%uDr5Ml(:8@Cb? << ;1i$#/oN&BK2B">iW)Z7,bVFC Thus point multiplication uses point addition and point doubling repeatedly to find the result. Q,Qs-MrMMI"@?99ps8q-I=(t0[ibGuX(m.l.&)5! Operations on Elliptic Curves. hVI7*(s[()7hbuQml3m-JpSQj5 >> UnYLnn-HH:5u9$5j,Tehe'R5B'I4k? S@jAt8ooU@RDHjT? All the operation such as addition, subtraction, division, multiplication involves polynomials of degree or less. ahY@;XDdW:GJr(*+R;<7Q[T, CWl=M#r.t4DrI\uPoNPAH!#/:! WTC\[AK2NF16N+gPCgjonjepG&kHFdhg_0pWkcnM0tb_\G=J9WU^k.EU,-Y0QdLdn[P6I6U'MC)E
For example, in the 1980s, elliptic curves started being used in cryptography and elliptic curve techniques were developed for factorization and primality testing. ,0nKfdQn/!/$Wuo+q]blGU2?rcigh(m]1E. endobj ]p1aP@C^h>Me27Te=s/!lj6>0>\C)>s,.4/T;"Xg>LBK*&O#g[YEIK 08dO.1SfNYVI0'5;jGMZcJ2)Mr endobj h7AI"jFRW5]_RUPD6u(hcUL5UBs+Z?E0j&qn72H]mTd'WrOO^MoWJg5R!JqC? 542 542 542 542 542 542 542 542 542 542 361 361 618 618 618 542 I9_LEi55R#^6/W85$EifQp5tCJK?r

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I've created this repository during the capture of the following online courses published on Udemy. ?e(r=F7PO !2X[9eRJ[F8+gS_ul4uRcoiXij0LLAk,r]Ird ] lSIHhGLZ]mUMoK5&a@S._#=DG$L*V1(+fS'\?AR6.'ZL-Nm;*\KfeE?2#=oGE1eTAL9c:! Work fast with our official CLI. /Parent 4 0 R (GS@m/(DO^*N";0\3%a&RnN+ga"i[V=WC^@e2@Bu8WgP9T,r;IM1oh._]ih83&^;pcUbNH(#"Xk+)s'Xbf(LMM4j(a4FuZf;XnLPQ'(EtU;'oT6OGfIGK%RRHICjS(I-jF0?%Tg81,(;FeClP ftZ56hdXShc/gUFN9BWh'XeocL6iQB@:obFCtk=G(f89&j/K%K? /Type /Page ]ji9B:PH(P=EY?mcFh9;[6g8TS8K7'YsR:1u7 /Contents 16 0 R s&12g_Mnst%ZmNU-tV^0+ugjh'>4"E_LI2rV]#^'_b&P8P[((_=LOk42m8% The addition rule is specified as follows: Rule to add the point at infinity to itself: Rule to add the point at infinity to any other point: Rule to add two points with the same x-coordinates when the points are either distinct or have y-coordinate 0: This also means that the negative of the point is. endstream rr@S!k\EF"oXR*?.ah#$hZ>]A[l5ctT'+t52nT5^"7Bm2u=Z.eNcU;+1I!;+1I!;+1I!;+1I!

stream ;cu>"APlW#qVdJj1P Consider two distinct points and so that and . /MediaBox [ 0 0 612 792 ] 72/Q!72/Q!72/Q!72/Q!72/Q!72/Q!72/Q!72/Q!? :,VM:&@I#bm3i(DU:@u5:,VM:&@I#bm3qu7K]?D=!20d%rrBd]J3-@RMuNd1 It was discovered by Whitfield Diffie and Martin Hellman in 1976, and uses a problem known as the Discrete Logarithm Problem (DLP) as its asymmetric operation.

N)$a?E.gn3R$\,@l'(p@ktr5-p3^ti4W$SmPCI_X8ZAo/t"![CZ:PsM5_=qAmC. 'Vm4la )&A(SCL&!<3$8>t#G.bPqRZ/#rULd*[.=!0@-[nT98t/#rUL To make operations on elliptic curve accurate and more efficient, the elliptic curve cryptography is defined over finite fields, also called Galois fields in honor of the founder of finite field theory, Évariste Galois. Point multiplication can be achieved by two basic elliptic curve operations, namely point addition and point doubling. EqfS6nkjMdqXm(q8c\-)U5Aq!U5Aq$"%p%WlV[0t2\;e4[\Y$C719_'/\be"onb&OVYbf'h6+Gi :KdbkWhjj;>M7k9g8$e*WaEm.-2LgRC2'R1Afbj-71+pr,*BN214^s9sTE:!!TE:!!TE:!!TE:!! !3BNB [ OWRp0!3g=\rekOFXBukF!6C5)J(Q[>rr>KqD,:7%'Mng!'Mng! h78juVT(Um7hn5ACTHOZYWKj-g'4pd2O$8:0[!0VT_pJ/7eNEegK"#js>8Lm/_:m=G9\j6%UBN_20HRJr2Be]o)(aaYrbk]nlr-+qG8?VDV68WZR3Q3:L=eg7

!L%+R#L63_I1NYBA2je-RXqADP\#XO'As/scLGGX='-PgutF;*eai8arH^qPoAJXm.qt GhU,@=YP&&UroHfLKmZ'h09dhSq+8Z\-h-,Isnq696P2LlsJf#'KO^Fpk1gsWABK`MW5\*C Even though the curve would no longer a gently flowing graph, as shown in the figure below, the algebraic equations for point addition and doubling still apply. 38 0 obj

[ endobj

S*r5I\t;a5o-h0WU-'F'$L)DL[q$^%%-(PqKq?r+Dn<7eb;50L/2]3&?#s(3!Ya.fZK$Z]6K*9g (?J2N"$9Z^d>u>*U4)tVK.L.2q If one knows and , one must guess at least the square root of the number of points on average to find .