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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 23 "Encryption & Decryption" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "To encrypt a message we need a
 method of turning a string of text into a number. We begin by orderin
g the alphabetic and the other key-board symbols, with " }{XPPEDIT 18 
0 "a;" "6#%\"aG" }{TEXT -1 13 " followed by " }{XPPEDIT 18 0 "b;" "6#%
\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b;" "6#%\"bG" }{TEXT -1 4 " by \+
" }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 6 ", etc:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "`crypt/a
lphabet` := \n   `abcdefghijklmnopqrstuvwxyz`\n   ||`ABCDEFGHIJKLMNOPQ
RSTUVWXYZ`\n   ||```1234567890-=~!@#$%^&*()_+`\n   ||` ,./<>?;':\"[]\{
\}|    `:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "The symbols || me
ans concatenate in Maple Version 6 and later versions. I am endebted t
o Professor John Cosgrave for the following function  (and its inverse
) that turns a string of text into a number. (See Professor Cosgrave's
 home page at: www.spd.dcu.ie/johnbcos/.)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "to_number := proc(s
t, string) \nlocal ll, nn, ss, ii;  \nll := length(st);\nif ll = 0 the
n RETURN(0) fi; \nnn := 1;\nfor ii to ll do \nss := SearchText(substri
ng(st, ii .. ii),\n                 `crypt/alphabet`);\nnn := 100*nn +
 ss \nod;\nnn - 10^(2*ll) \nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 85 "Next we do some examples, where the text needs to be included w
ithin backward quotes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "N:=to_number(`I came I saw I conque
red.`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"S\"[]!=0@<9:.zNzB,>
zNz08,.zN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 191 "The initial 35 represents the letter `I', the  79 is the
 blank, the 03 is the letter `c', and 81 is the period at the end.  Ne
xt we need a program to recover a string of text from the number." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "from_number := proc(nn, int
eger) \nlocal ss, mm, ll, pp, ii, ans; mm := nn;\nll := floor(1/2*trun
c(evalf(log10(mm))))+1;\nans := ``; for ii to ll do mm := mm/100;\npp \+
:= 100*frac(mm);\nss := substring(`crypt/alphabet`, pp..pp);\nans := c
at(ss, ans); mm := trunc(mm)\nod; ans end:\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "from_number(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#%:I~came~I~saw~I~conquered.G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 93 "It should be noted that if we use th
e above function on a randomly chosen number we will get " }{TEXT 256 
12 "gobbledegook" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "R:=rand(10^85)();" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"RG\"`p4V(3h_WZb\"Q;*fz3r`N6Re`2sccad!>8W
QWUQk?il8[2$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "from_number
(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KcgVm=9t-LPRLRmse(S=!t(I<]m2K
j'~pLo1*Szj'QiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 135 "Next we generate public and private keys for our fr
iends Bob and Alice who wish to communicate privately. First for Alice
 and then Bob." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "p[a]:=nextprime(rand(10^60)());" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%\"pG6#%\"aG\"gnTQz\")o6xsc)*=*Rh)yS_#GfvO0*>#
Q3Ba-&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q[a]:=nextprime(r
and(10^70)());" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"qG6#%\"aG\"`o\"
G1\\JyJ#fq%f,M+%)4XO([0()z%H;E,VR&zN$Gx!)" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "n[a]:=p[a]*q[a]; # " }{XPPEDIT 18 0 "n[a];" "6#&%\"
nG6#%\"aG" }{TEXT -1 23 " is public but not the " }{XPPEDIT 18 0 "p[a]
;" "6#&%\"pG6#%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q[a];" "6#&%
\"qG6#%\"aG" }{TEXT -1 1 "." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG
6#%\"aG\"\\s@`,yIr%>`xP1m8&><:I[A(3At,e4W#)G[`k*[*>S_f!G\\io'oB7Su#ewi
x-%\\t%HjTiw\"fS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "length(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$H\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "e[a]:=nextprime(12345678987654321);# " }
{XPPEDIT 18 0 "e[a];" "6#&%\"eG6#%\"aG" }{TEXT -1 16 " is also public.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"eG6#%\"aG\"2tVl()*ycM7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "phi_na:=(p[a]-1)*(q[a]-1); #
" }{TEXT -1 11 "This number" }{MPLTEXT 1 0 1 " " }{XPPEDIT 18 0 "phi(n
[a]);" "6#-%$phiG6#&%\"nG6#%\"aG" }{TEXT -1 16 " is kept secret." }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'phi_naG\"\\s+_J\")yT%**zW)G@>DUD)Rn
5EtoWj$**z()3`)>O79>S_f!G\\io'oB7Su#ewix-%\\t%HjTiw\"fS" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "igcd(e[a],phi_na);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "d[a]:=e[a]^(-1) mod phi_na; # " }{XPPEDIT 18 0 "d[a];" "6#&%\"dG6#
%\"aG" }{TEXT -1 23 " is very, very private." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"dG6#%\"aG\"\\sP!pM!QhAf$GfM_$pt2K)H$G=:`;Ld&e7spY^
v,7!)3i$4\"p+BJ(H-3B(HN$p!34(Q+N@y$\\AP" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "e[a]*d[a] mod phi_na;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p[b]:=nextp
rime(rand(10^58)());" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#%\"bG
\"en^86.23d+TM!pxBX25a%*Gi[J\"Gtp8D[" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "q[b]:=nextprime(rand(10^68)());" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"qG6#%\"bG\"_op8)zrDv]FU&G\"f0G[K?c^U_^\"*)G*47#Hb%
f9r#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "n[b]:=p[b]*q[b];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"nG6#%\"bG\"ir>&>/NX'znVnnB&otu%*
Ge\"yuca7*G))H\"Qt)[P#*z,99!)f#zAZlse=1aNcykhtA$GpK;$38" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "length(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"$E\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "e[
b]:=nextprime(987654321234567);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"eG6#%\"bG\"0(eM7Kaw)*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "phi_nb:=(p[b]-1)*(q[b]-1);" }{TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'phi_nbG\"ir+oKH*Q]@*zy[b\"R+>_a2,Vmi_pQTm
1wT*G7s<ST,)f#zAZlse=1aNcykhtA$GpK;$38" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "d[b]:=e[b]^(-1) mod phi_nb;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%\"dG6#%\"bG\"hrB23XObFxIl\">s\"\\;mLM)=B*4Ixjt\"f7P.
*))pAlJL^Kqnbm%QQ<8j%o`ktaM-(pM#)34?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Now we are ready for Bob to sen
d a message to Alice. He first uses his secret exponent " }{XPPEDIT 
18 0 "d[b];" "6#&%\"dG6#%\"bG" }{TEXT -1 0 "" }{TEXT -1 58 " to sign h
is message before using Alice's public exponent " }{XPPEDIT 18 0 "e[a]
;" "6#&%\"eG6#%\"aG" }{TEXT -1 43 " to encrypt the message. First the \+
message " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "M:=to_
number(`Meet me at Fog City at 8:00pm. I am madly in love with you.`);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"ar\"=_^#z3?4Bz0A:7z94zD7/,
8z8,zNz\"QhJO')='z?,zD?4Hz2:Kz?,z08z?00R" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "S:=M&^d[b] mod n[b];" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"SG\"hrqxe#fAKn8/V.SWGeU!e(=VmhwqJt(Q*e!)RF$>!ou\"32.B*3@\"*fc?n
HWv9HRX2Sa)*G&R*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C:=S&^e
[a] mod n[a];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG\"[s$p!)f***pE*
ogC)p5$pO)f)puMe=*e?\"z!*pH;;rNT^VA>Jeq)GZoIX'**y$G5Ty!)H/meW\\]bl" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 27 "Alice receives the message " }{XPPEDIT 18 0 "C;" "6#%\"CG" }
{TEXT -1 39 " from Bob and uses her secret exponent " }{XPPEDIT 18 0 "
d[a];" "6#&%\"dG6#%\"aG" }{TEXT -1 156 "  to decrypt. Then when she re
alizes that the message does not make sense and that the message is su
pposed to come from Bob, she uses Bob's public exponent " }{XPPEDIT 
18 0 "e[b];" "6#&%\"eG6#%\"bG" }{TEXT -1 1 " " }{TEXT -1 24 "to retrie
ve the message " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "C&^d[a] mod n[a];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"hrqxe#fAK
n8/V.SWGeU!e(=VmhwqJt(Q*e!)RF$>!ou\"32.B*3@\"*fc?nHWv9HRX2Sa)*G&R*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "%&^e[b] mod n[b];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"ar\"=_^#z3?4Bz0A:7z94zD7/,8z8,zNz\"QhJO')=
'z?,zD?4Hz2:Kz?,z08z?00R" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"from_number(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fnMeet~me~at~Fog~
City~at~8:00pm.~I~am~madly~in~love~with~you.G" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "So Alice is now sure t
his message comes from Bob. But, will she meet him???" }}}}{MARK "3 0 \+
0" 267 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }