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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 23 "MAT312: Applied Algebra" }}
{PARA 18 "" 0 "" {TEXT -1 19 "Computer project #1" }}{PARA 19 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 19 8 "Message " }{TEXT 18 239 "b
elow is a message which has been encrypted using the RSA public key al
gorithm.  In its encrypted form, it  consists of  12 numbers of about \+
210 decimal digits each.   To get Maple to read it, you must hit the e
nter key for each statement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2553 "Message :=[\n2506687366489
7638140085303971169651970027323900993076899383071677609322867743924029
5257178981316117701965624531104681458002208123032739455036441595688900
759757446466769583605417595786288394551208695477810891237, 24784968265
5655471804778511653871068180575500506496445672016587529503240181045730
2506157539950880634253546180590506024796659396683413536962166327024423
84515847788360895757550652478657543926477631892150313607981, 158888150
6235651965103917827362855488493455874440804424153506787724334638736063
8182465878743390528332866873169227109134388368844477059275432437081228
8411526113824574066229853220150124534146420063066674871855199, 1171312
0866445676943254732114223162365704806282768849831417770173605433053267
5519552661766304185661524609961916813128312161514809099995726998115418
381951863530436556157042842660730462932805835224547159284816740, 26712
0648655711195114497528395544897196016955040936469542380483686533524849
8352694386098392888868002540972045337279551168740642502811725197304531
70348333981151651248228551415519215212586190598658444540481109856, 677
7906472735566816243836385657453513205134744604240536631988497498214533
6362396676349416077111575671613221425354563748712888678352156325550591
90354761033008989984565322669665188828297490724841391423164185633, 265
0605063384209761477025434619955419838813369037038035145177010312321054
2796604854876783562644860459915128778567766115967109415085972370725447
6140521486489354114718637027760701307547925715802976542431336473443, 2
1092557005712995912983012607225038468597999935512627981724216475262099
0494825428163700890823013307942267277918208857036332759955422909551928
768341433501401849297988437512790988023406088121035312585410317782475,
 658669531488053063427327216533844273461060489685756364391651542910487
1285747419778037063268509510499966434028880834117396301356882661652883
7758313014516195657318311016628854261310963745162078100050716226611115
, 16036480732165435511717302450423177727375044892129676032240851923055
9210201182897852667639536616716633724735675259319287829292309149745538
2030361917121196627480268161069762959767156228230337172868444154864238
24, 226348620153385254622143996318532553026502038858722559629681446428
2169925574989395098213380935584065707345125666089413766583060587137440
2215308157914746451446315439926583558026584604175233506754955057517713
0890, 9765920028938720280000088416637335196913605308565795158492543170
2078703423201001205142438546998319164792543498737530279138080277711254
6417109863078342753713824010093251498117508837748016250276335043947243
86126\n]:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 69 "To reference
 any one of the numbers in the message, use the notation " }{TEXT 19 
10 "Message[n]" }{TEXT -1 9 " for the " }{XPPEDIT 18 0 "n^th;" "6#)%\"
nG%#thG" }{TEXT -1 20 " number in the list." }}{PARA 0 "" 0 "" {TEXT 
-1 73 "(note that the : at the end of the statement supresses any Mapl
e output.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "The message was encoded using the following " }{TEXT 19 4 "base
" }{TEXT -1 74 ", which happens to be the product of two primes of abo
ut 100 digits each. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "bas
e := 30645237836843904806816346282773271851874635743981678646285352705
1887614619645017555473997034658600670196373847577090557186847481568857
5148639434207680791924965821038451065702409095292295135390679933744604
89159;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
137 "The message was encoded using the exponent below, which will be o
f little help in deciphering the message.  However, you could use it t
o " }}{PARA 0 "" 0 "" {TEXT -1 22 "forge a different one." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "encodingExponent:=1872782342363042
6394917883250872196001257120579902357744184497884282647216527538382593
9373;" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The exponent  " }{TEXT 19 1 "x
" }{TEXT -1 63 "  below will allow you to decipher the message, by com
puting , " }{XPPEDIT 19 1 "a^x;" "6#)%\"aG%\"xG" }{TEXT -1 5 " mod " }
{XPPEDIT 19 1 "base;" "6#%%baseG" }{TEXT -1 9 ", where  " }{XPPEDIT 
19 1 "a;" "6#%\"aG" }{TEXT -1 45 " represents each number in the messa
ge above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 215 "x := 98246007475185045209994421243798892261304114349
0535054203552630969073441462583645630894836103108240682747507658949699
7390708757122628367151859091212055673903930387563218530109465392784229
4310968495586626369445;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 586 "Of course, the resulting computation gives you nu
mbers, and you would like something you can read.  To do so, you will \+
need to know how the words were converted to large integers.  Here's h
ow:  the message was read in 47 characters at a time.  Each character \+
was viewed as a number between 0 and 127 (by assigning it the correspo
nding ASCII code), and the resulting block was interpreted as a 47-dig
it number in base 128  (such numbers are about 210 digits long when wr
itten in base 10). Spaces, punctuation, line breaks, and so on were tr
eated as part of the message, and also encoded." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "For example, to encode th
e first 47 characters of the previous paragraph,  we would compute " }
}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "79*128^46+102*128^45
+32*128^44+99*128^43+111*128^42;" "6#,,*&\"#z\"\"\"*$\"$G\"\"#YF&F&*&
\"$-\"F&*$\"$G\"\"#XF&F&*&\"#KF&*$\"$G\"\"#WF&F&*&\"#**F&*$\"$G\"\"#VF
&F&*&\"$6\"F&*$\"$G\"\"#UF&F&" }{TEXT -1 8 "+ ... + " }{XPPEDIT 18 0 "
32*128^3+121*128^2+111*128+117;" "6#,**&\"#K\"\"\"*$\"$G\"\"\"$F&F&*&
\"$@\"F&*$\"$G\"\"\"#F&F&*&\"$6\"F&\"$G\"F&F&\"$<\"F&" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 246 "because O=79, f=102, space=32, c=99,
 o=111, (and so on) in the ASCII code.  To write such a number as prin
table text, we just reverse the process.  Rather than make you worry h
ow to do all that,  below is a little Maple procedure which does that:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 165 "PrintNumberAsText:= proc(num)\n  local i,p,k;\n  p:=128;\n  k
:=47;\n  printf(\"%s\",\n         convert([seq(  iquo(modp(num,p^(k-i+
1)), p^(k-i)), i=1..k)],'bytes'));\n  end;" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Make sure that you can use " }
{TEXT 19 17 "PrintNumberAsText" }{TEXT -1 1 " " }{TEXT 19 0 "" }{TEXT 
-1 142 "to decipher what the number below represents.  (Note that this
 process, while hard for people to read, is NOT encryption, merely tra
nslation)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 111 "trialnum:= 767881737176172013294653540993500170940
865317690485262301912883980080798615353381549606267397341184;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "Finally, you might need to know
 how to compute powers modulo p in Maple-- you do this using " }{TEXT 
19 2 "&^" }{TEXT -1 5 " and " }{TEXT 19 3 "mod" }{TEXT -1 26 ".  For e
xample, to compute" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`
mod`(5^2,4);" "6#-%$modG6$*$\"\"&\"\"#\"\"%" }{TEXT -1 21 ", we do the
 following" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "5 &^ 2 mod 4;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "A more complicated example mig
ht be to compute the third number in the message raised to the 178th p
ower, mod 47:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Message[3] \+
&^ 178 mod 47;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 106 "or, if you prefer to write modular arith
metic in functional notation, you could write the same command as " }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "modp(Message[3] &^ 178, 47)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "That should be enough informati
on to do the project." }}{PARA 0 "" 0 "" {TEXT -1 50 "You are now fina
lly ready to decipher the message!" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "36 0" 2 }{VIEWOPTS 1 1 0 3 2 1804 }