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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 224 "1. Decode the following
 phrase, which was encoded using a affine encoding cipher on a 27-lett
er alphabet (the letters a-z and a blank):\n   message = \"xgb dkjvmbk
ffcmtkrv m lkaerask mdbmwkfmezremzkmxrcmkkmfvkrx mxgvkmaskrvsc\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "message := \"xgb dkjvmbkffcm
tkrv m lkaerask mdbmwkfmezremzkmxrcmkkmfvkrx mxgvkmaskrvsc\";" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(messageGQdoxgb~dkjvmbkffcmtkrv~m~lk
aerask~mdbmwkfmezremzkmxrcmkkmfvkrx~mxgvkmaskrvsc6\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 145 "First, find the most commonly occuring charact
ers.  This is much easier if you arrange that all copies of the same c
haracter be grouped together." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "nums := convert(message,bytes);" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%%numsG7eo\"$?\"\"$.\"\"#)*\"#K\"$+\"\"$2\"\"$1\"\"$=\"\"$4\"F(F+
\"$-\"F/\"#**F.\"$;\"F+\"$9\"F-F)F.F)\"$3\"F+\"#(*\"$,\"F2F4\"$:\"F+F)
F.F*F(F.\"$>\"F+F/F.F5\"$A\"F2F5F.F8F+F.F&F2F0F.F+F+F.F/F-F+F2F&F)F.F&
F'F-F+F.F4F6F+F2F-F6F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "s
ort(nums);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7eo\"#KF$F$F$F$\"#(*F%F%
\"#)*F&F&\"#**F'F'\"$+\"F(\"$,\"F)F)\"$-\"F*F*F*\"$.\"F+\"$1\"\"$2\"F-
F-F-F-F-F-F-F-F-F-F-\"$3\"\"$4\"F/F/F/F/F/F/F/F/F/F/F/\"$9\"F0F0F0F0F0
\"$:\"F1F1\"$;\"\"$=\"F3F3F3F3\"$>\"\"$?\"F5F5F5\"$A\"F6" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "convert(%,bytes);" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#Qdo~~~~~aaabbbcccddeeeffffggjkkkkkkkkkkkklmmmmmmmmmm
mmrrrrrrssstvvvvvwxxxxzz6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Co
nclusion: 'm' is the most common, followed by 'k'." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 40 "Alphabet:=\"abcdefghijklmnopqrstuvwxyz \";
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)AlphabetGQ<abcdefghijklmnopqrst
uvwxyz~6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "ToNum:=proc \+
(text)\n  local i;\n  global Alphabet;\n\n  [seq(SearchText(substring(
text,i),Alphabet)-1,i=1..length(text))];\nend:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 123 "FromNum:=proc(numlist)\n  local i;\n  global \+
Alhabet;\n\n  cat(seq(substring(Alphabet,numlist[i]+1), i=1..nops(numl
ist)));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "AffineDec
ode:=proc(plaintext,a,b)\n  local textnum,codenum,i,p;\n  global Alpha
bet;\n\n  p:=length(Alphabet);\n  textnum:=ToNum(plaintext);\n  codenu
m:=[seq(((textnum[i]-b)/a) mod p,i=1..length(plaintext))];\n  FromNum(
codenum);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Guess that ' '
 was encoded to 'm' and 'e' to 'k'." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "ToNum(\" mek\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7
&\"#E\"#7\"\"%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "msolv
e(\{a*26+b=12,a*4+b=10\},length(Alphabet));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"bG\"#</%\"aG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 96 "AffineDecode(\"xgb dkjvmbkffcmtkrv m lkaerask mdbmwkf
mezremzkmxrcm kkmfvkrx mxgvkmaskrvsc\",5,17);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#Qeomonsieur~neddy~wears~spectacles~in~bed~that~he~may~s
ee~dreams~more~clearly6\"" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 371 "2
. Sometime soon, you will be sent e-mail containing a message encoded \+
by an affine matrix cipher,\nbased on 2-vectors in the 61-character\n \+
   alphabet = \"ABCDEFGHIJKLMNOPQRSTUVWXYZ\\nabcdefghijklmnopqrstuvwxy
z.,:;!?' \",\nalong with the first few letters of the message. Decrypt
 the message.\n    message = \"?PV;gwENokHVV;b.I?IUWYBlFjokmfq:ZV\".\n
    First chars = \"Send roses\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"message := \"?PV;gwENokHVV;b.I?IUWYBlFjokmfq:ZV\";" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%(messageGQC?PV;gwENokHVV;b.I?IUWYBlFjokmfq:ZV6\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Alphabet := \"ABCDEFGHIJKL
MNOPQRSTUVWXYZ\\nabcdefghijklmnopqrstuvwxyz.,:;!?' \";" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%)AlphabetGQhnABCDEFGHIJKLMNOPQRSTUVWXYZ|+abcdefg
hijklmnopqrstuvwxyz.,:;!?'~6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 123 "FromNum:=proc(numlist)\n  local i;\n  global Alhabet;\n\n  ca
t(seq(substring(Alphabet,numlist[i]+1), i=1..nops(numlist)));\nend:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "ToNum:=proc (text)\n  loc
al i;\n  global Alphabet;\n\n  [seq(SearchText(substring(text,i),Alpha
bet)-1,i=1..length(text))];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 218 "TextToVects := proc (text,n)\n  local i, j, textnums
, vectors;\n\n  textnums := ToNum (cat (text, seq(` `, i=1..modp (n - \+
length(text), n))));\n  [seq ( [seq (textnums [(j-1)*n + i], i=1..n)],
 j=1..nops(textnums)/n)];\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "VectsToText := vects -> FromNum (map(op,map(convert,vects,list
))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "AffineMatEncode :=
 proc (text, A, B)\n  local vtext, vcrypt, i, n, p;\n  global Alphabet
;\n\n  p := length (Alphabet);\n  n := nops (convert(B,list));\n\n  vt
ext := TextToVects (text, n);\n  vcrypt := [seq (map (modp, evalm (A &
* vtext[i] + B), p), i=1..nops(vtext))];\n  VectsToText (vcrypt);\nend
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A := matrix(2,2, [[a,b
],[c,d]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%
\"aG%\"bG7$%\"cG%\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "B \+
:= array(1..2, [e,f]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'vec
torG6#7$%\"eG%\"fG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "crypt
 := TextToVects(message,2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&cryp
tG737$\"#e\"#:7$\"#@\"#c7$\"#L\"#\\7$\"\"%\"#87$\"#T\"#P7$\"\"(F*F)7$
\"#G\"#`7$\"\")F'7$F;\"#?7$\"#A\"#C7$\"\"\"\"#Q7$\"\"&\"#OF27$\"#R\"#K
7$\"#V\"#b7$\"#DF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plain
 := TextToVects(\"Send roses\",2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%&plainG7'7$\"#=\"#J7$\"#S\"#I7$\"#g\"#W7$\"#T\"#X7$F(F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq1 := evalm (A &* crypt[1] +B - pl
ain[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G-%'vectorG6#7$,*%\"
aG\"#e%\"bG\"#:%\"eG\"\"\"!#=F/,*%\"cGF+%\"dGF-%\"fGF/!#JF/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq2 := evalm (A &* crypt[2] \+
+B - plain[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G-%'vectorG6#
7$,*%\"aG\"#@%\"bG\"#c%\"eG\"\"\"!#SF/,*%\"cGF+%\"dGF-%\"fGF/!#IF/" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eq3 := evalm (A &* crypt[3]
 +B - plain[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G-%'vectorG6
#7$,*%\"aG\"#L%\"bG\"#\\%\"eG\"\"\"!#gF/,*%\"cGF+%\"dGF-%\"fGF/!#WF/" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "msolve(\{eq1[1]=0,eq1[2]=
0,eq2[1]=0,eq2[2]=0,eq3[1]=0,eq3[2]=0\},length(Alphabet));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<(/%\"bG\"\"$/%\"aG\"#R/%\"dG\"#:/%\"cGF,/%
\"eG\"#H/%\"fG\"#M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(
%,\{evalm(A),evalm(B)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'vect
orG6#7$\"#H\"#M-%'matrixG6#7$7$\"#R\"\"$7$\"#:F2" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 57 "AffineMatEncode (message, [[39, 3], [15, 15]]
, [29,34]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QCSend~roses~and~cavia
r.~James~Bond.6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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