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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Mathematik f\374r Informat
iker IV (Algebra)" }}{PARA 0 "" 0 "" {TEXT -1 15 "Aufgabenblatt 1" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Als erstes schre
iben wir eine kleine Prozedur, die das k-te Bit einer Zahl (von rechts
) ermittelt. " }}{PARA 0 "" 0 "" {TEXT -1 85 "Idee: Um zum Beispiel da
s 6te Bit von 1010101010 zu erhalten, schieben wir diese Zahl" }}
{PARA 0 "" 0 "" {TEXT -1 86 "           5 mal nach rechts, erhalten da
nn 10101 und schauen uns das unterste Bit an." }}{PARA 0 "" 0 "" 
{TEXT -1 112 "           Nach rechts schieben entspricht durch 2 teile
n und abrunden, und das unterste Bit ist das Vorzeichen." }}{PARA 0 "
" 0 "" {TEXT -1 14 "Als Prozedur: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "bin_test:=proc(x,k)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "  local d:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "  d:=iquo(x,2^(
k-1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "  if type(d,even) then re
turn(0): elif type(d,odd) then return(1) fi:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Aufgabe 1
 (a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "a:=rand(10^99..10^100-1)();" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"aG\"_q4V(3h_WZb\"Q;*fz3r`N6Re`2sccad!>8WQWUQk?il8[2
8\\mie+QN" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 42 "Ermittle Anzahl der
 bin\344ren Stellen von a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "bin_stellen:=floor(log[2](a))+1;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,bin_stellenG\"$J$" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 145 "Um das k-te Bit zu ver\344ndern, ermitte
ln wir erst den Wert dieses Bits. Ist dieser 0, so addieren wir 2^(k-1
), ansonsten ziehen wir diese Zahl ab." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "k:=rand(0..bin_stellen)
(): k1:=rand(0..bin_stellen)(): k2:=rand(0..bin_stellen)():" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "if bin_test(a,k)=0 then b1:=
a+2^(k-1): else b1:=a-2^(k-1): fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#b1G\"_q\"e!))GPiDcd88hS;,[,4MAmafL3ij+ZzqUwEOk?il8[28\\mie+QN" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "if bin_test(a,k1)=0 then b2:
=a+2^(k1-1): else b2:=a-2^(k1-1): fi:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "if bin_test(b2,k2)=0 then b2:=b2+2^(k2-1): else b2:=b
2-2^(k2-1): fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G\"_qHMyE\"y/M
_bFZ*f$p=U%3nX'e7f%z7(3dq\"y5\"=6kRV__59I\"\\mie+QN" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "p:=nextprime(rand(10^9..10^10-1)());" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"+B')ySs" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "b1_fp:=b1 mod p; b2_fp:=b2 mod p;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&b1_fpG\"+JE\"='o" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&b2_fpG\"+.sz8Z" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
111 "Den folgenden Befehl auf eigenes Risiko ausf\374hren! (kann unter
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" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ifactor(b
1-b2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,)-%!G6#\"\"#\"$F#\"\"\"-F&
6#\"$,\"F*-F&6#\"&6*pF*-F&6#\"%x#*F*-F&6#\"%(H%F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "Die Zertifikate u
nterscheiden sich deutlich. Im Folgenden wird eine obere Schranke f
\374r die Wahrscheinlichkeit berechnet, dass diese Zertifikate " }}
{PARA 0 "" 0 "" {TEXT -1 133 "doch gleich sind. Die Formel daf\374r is
t N*ln(t)/t, wobei N die Anzahl Bit der Zahl a und t eine obere Schran
ke f\374r die Primzahl p ist." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "N:=floor(log[2](10^100))+1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"$L$" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "wkeit:=evalf(N*ln(10^10)/10^10);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&wkeitG$\"+g$3wm(!#;" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 13 "Aufgabe 1 (b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "a:=rand(10^29..10^30-1)();" 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "if bin_test(a,k)=0 then b1:=
a+2^(k-1): else b1:=a-2^(k-1): fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "if bin_test(b2,k2)=0 th
en b2:=b2+2^(k2-1): else b2:=b2-2^(k2-1): fi;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#b2G\"?**\\V;N()f$)yqknt%4\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "count1:=0: p:=2: count2:=0: p1:=[]: p2:=[]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "while p<10^3 do" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 66 "  if a mod p = b1 mod p then count1:=count1+
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d p = b2 mod p then count2:=count2+1: p2:=[op(p2),p]: fi:" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "  p:=nextprime(p):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "coun
t1; count2; p1; p2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#7#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"#\"\"(\"$F\"\"$P$
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Alternativ dazu, und vor alle
m auch einfacher:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "ifactor(a-b1); ifactor(b2-a);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*$)-%!G6#\"\"#\"#9\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#**)-%!G6#\"\"#\"#B\"\"\")-F&6#\"\"(F(F*-F&6#\"$F\"F*-F&
6#\"$P$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Wie man sieht, gibt
 es nur eine Primzahl, die a-b1 teilt: 2. Das ist nicht verwunderlich,
 da a ja nur um eine Zweierpotenz ver\344ndert wurde." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Aufgabe 1 (c)
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{MPLTEXT 1 0 15 "N:=bin_stellen;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"NG\"#(*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "wkeit:=evalf(N
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!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Z\344hle hier die Anzahl d
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0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"count;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "evalf(4/168);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+\"Q_4Q#!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Anhang: D
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