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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 211 33 "Implementation of  RSA a
lgorithm." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 77 "Initialize from rea
ding the necessary commands from the publisher's web-site:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "with(LinearAlgebra): with(linalg): \+
with(numtheory): with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "n
umb:=table([\" \"=0,\"a\"=1, \"b\"=2, \"c\"=3, \"d\"=4, \"e\"=5, \"f\"
=6, \"g\"=7, \"h\"=8, \"i\"=9, \"j\"=10, \"k\"=11, \"l\"=12, \"m\"=13,
 \"n\"=14, \"o\"=15, \"p\"=16, \"q\"=17, \"r\"=18, \"s\"=19, \"t\"=20,
 \"u\"=21, \"v\"=22, \"w\"=23, \"x\"=24, \"y\"=25, \"z\"=26]):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "alph:=table([0=\" \",1=\"a\",2=\"b
\",3=\"c\",4=\"d\",5=\"e\",6=\"f\",7=\"g\",8=\"h\",9=\"i\",10=\"j\",11
=\"k\",12=\"l\",13=\"m\",14=\"n\",15=\"o\",16=\"p\",17=\"q\",18=\"r\",
19=\"s\",20=\"t\",21=\"u\",22=\"v\",23=\"w\",24=\"x\",25=\"y\",26=\"z
\"]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "text2num:=proc(txt)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  local i,z; z:=0;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "  for i from 1 while i<=length(txt) do" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "    z:=100*z+numb[substring(txt,i)];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "  return(z);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "num2text:=proc(n)" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "  local i,w,z; z:=NULL;w:=n;" }}{PARA 0 "> "
 0 "" {MPLTEXT 1 0 35 "  for i from 1 while 100^(i-1)<n do" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 30 "    z:=cat(alph[w mod 100],z);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "    w:=floor(w/100);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  retu
rn(z);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 208 
"" 0 "" {TEXT 212 27 "BEGINNING OF IMPLEMENTATION" }}{PARA 209 "" 0 ""
 {TEXT 207 143 "Suppose Alice wants to transmit securely a message \"H
ello world\". She first converts it into a number, which will play a r
ole of her plaintext:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "m:
=text2num(\"helloworld\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"4/7=:B:7
70)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 63 "To form his private and p
ublic key, Bob chooses 2 large primes:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "p:=nextprime(rand(10^20..10^24)());" }}{PARA 11 "" 1 
"" {XPPMATH 20 "\"9F3L\"4Wq.eYW0%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "q:=nextprime(rand(10^20..10^24)());" }{MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"9H)oH/cE\\Q-%R**" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 207 51 "So that the exponentiation will be made \+
modulo n=pq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=p*q;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"Q$e\"z(omB8HF8*4O_#y(H\"\\5gx)HS" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 207 40 "Then Bob chooses the encryption e
xpoent:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "e:=53;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "\"#`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 60 "
He has to make sure that e is coprime with phi(n)=(p-1)(q-1)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gcd(e,(p-1)*(q-1));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
207 39 "So that he can compute the secret key d" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 26 "d:=e^(-1) mod (p-1)*(q-1);" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "\"Qd.7KA'R66t6I8l[;'QL_'RJ09\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 207 58 "Now Bob publishes (n,e) and keeps (p,q,d) as a secret \+
key." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 56 "Using Bob's public key, \+
Alice encrypts her message m as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 13 "c:=m^e mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"P)>uia$H[K49
l*)ek!eZ<X!)pI=Y" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 207 98 "To decrypt t
his message, Bob raises it to the secret exponent d and converts the r
esult into text:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "m_bob:=
(c&^d) mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"4/7=:B:770)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 207 13 "and converts " }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 16 "num2text(m_bob);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "Q+helloworld6\"" }}}}
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