{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 36 "L\366sung zu Probeklausur Februar 2009" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Aufgabe 1 (Secret Sharing)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "p:=211; # any prime number larger than the largest possible secr et value\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"$6#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x:=[1,2,3];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=[131,149,177];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"xG7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG7% \"$J\"\"$\\\"\"$x\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "sec ret:=0:\nfor i from 1 to nops(x) do\n s:=f[i]:\n for j from 1 to n ops(x) do\n if (j<>i) then s:=modp(s*(-x[j]/(x[i]-x[j])),p): fi: \+ # Lagrange interpolation formula\n od:\n secret := secret + s mod p:\nod;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"$J\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'secretG\"$#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"$\\\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sec retG\"$d\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"$x\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'secretG\"$B\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "'Zu Fu\337' kommt man mit Lagrange-Interpolation auf folg ende Formel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "3*131-3*149 +177 mod 211;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$B\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a0:=secret; a1:='a1'; a2:='a2';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G\"$B\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1GF $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2GF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "fo r i from 1 to nops(x) do eq[i] := f[i]=a0+a1*x[i]+a2*x[i]*x[i] mod p; \+ od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"\"/\"$J\",(\"$B \"F'%#a1GF'%#a2GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"#/ \"$\\\",(\"$B\"\"\"\"*&F'F,%#a1GF,F,*&\"\"%F,%#a2GF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"$/\"$x\",(\"$B\"\"\"\"*&F'F,%#a1GF,F, *&\"\"*F,%#a2GF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "solve (\{eq[1],eq[2],eq[3]\},[a1,a2]); assign(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#7$/%#a1G\"\"$/%#a2G\"\"&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "f10:=a0+a1*10+a2*10*10 mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f10G\"#?" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" } }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 "Aufgabe 2 (Diophantische Gleichung und Algo Erw. Euklid) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x:='x': y:='y':\nisolve(75*x+20*y=4 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Also keine L\366sung f\374r die 1. Gleichung." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "isolv e(27*x-59*y=-10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG,&\"\"%\" \"\"*&\"#fF(%$_Z1GF(F(/%\"yG,&\"\"#F(*&\"#FF(F+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Also: " }{TEXT 260 9 "Spezielle" }{TEXT -1 105 " L\366sung (x,y) = (4,2) (Mit Algo Erweiterter Euklid kommt (x,y)=(240 ,110) heraus, was auch richtig ist.). " }{TEXT 259 15 "\nAlle positive n" }{TEXT -1 67 " L\366sungen: S\344mtliche (x,y) nach obiger Maple-Gl eichung f\374r _Z1 >= 0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 26 "Aufgabe 3 (Potenzen mod m)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "121^38 mod 11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "(26*26)^20 mod 5;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "1235^39 mod 9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \")" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Aufgabe 4 (Multiple Choice)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "4.1. nur c) ist richtig" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "4.2. a)b)c) sind richtig" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "10 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }