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{SECT 0 {EXCHG {PARA 260 "" 0 "" {TEXT -1 10 "Lecture 12" }}{PARA 19 "
" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 39 "The RSA \+
Public Key Cryptographic System" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "
The " }{TEXT 328 10 "RSA system" }{TEXT -1 6 " is a " }{TEXT 329 31 "p
ublic-key cryptographic method" }{TEXT -1 78 " that permits both encry
ption and digital signatures (authentication). Ronald " }{TEXT 351 1 "
R" }{TEXT -1 11 "ivest, Adi " }{TEXT 352 1 "S" }{TEXT -1 19 "hamir, an
d Leonard " }{TEXT 353 1 "A" }{TEXT -1 153 "dleman developed the RSA s
ystem in 1977.  RSA stands for the first letter in each of its invento
rs' last names.  For more informations you may start with" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{TEXT 
284 46 "http://www.rsasecurity.com/rsalabs/faq/3.html " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "or you may do a web \+
luis search at the USF library on the key word " }{TEXT 285 12 "crypto
graphy" }{TEXT -1 34 " for local books on the subject.  " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 36 "Here's the way t
he RSA system works:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 155 "Suppose Alice wants Bob to send her a secret m
essage using the RSA system. Here's what she does. [The communication \+
can take place via email, for example.]" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 38 "1. She finds two large primes p and \+
q." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "2. \+
She forms the product m = pq. We call m the " }{TEXT 286 7 "modulus" }
{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 16 "3. She computes " }
{XPPEDIT 18 0 "phi(m) = (p-1)*(q-1);" "6#/-%$phiG6#%\"mG*&,&%\"pG\"\"
\"F+!\"\"F+,&%\"qGF+F+F,F+" }{TEXT -1 25 ". [This is the so-called " }
{TEXT 331 18 "Euler phi function" }{TEXT -1 28 " applied to the modulu
s m. ]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 
"4. She chooses a positive integer e such that igcd(e," }{XPPEDIT 18 
0 "phi(m);" "6#-%$phiG6#%\"mG" }{TEXT -1 6 ") = 1." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "5. She computes the " }
{TEXT 287 7 "inverse" }{TEXT -1 18 ", call it d, of e " }{TEXT 288 6 "
modulo" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG" }
{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 92 "6. She sends the pair \+
of numbers [m,e] to Bob. This pair of numbers may be made public, but \+
" }{TEXT 257 11 "the numbers" }{TEXT -1 8 " p,q, d " }{TEXT 259 3 "and
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG" }{TEXT 
-1 1 " " }{TEXT 258 23 "should be kept secret! " }{TEXT -1 19 " We cal
l [m,e] the " }{TEXT 261 10 "public key" }{TEXT -1 12 " and d, the " }
{TEXT 262 11 "private key" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "7. She tells Bob how to use Ma
ple to convert the text of his message into a sequence of positive int
egers " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 10 ", . . . , " }
{XPPEDIT 18 0 "a[k];" "6#&%\"aG6#%\"kG" }{TEXT -1 51 " each less than \+
m. One such method is shown below. " }{TEXT 260 31 "This method can be
 made public." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "8. She tells Bob to compute for each i the number " }
{XPPEDIT 18 0 "b[i] = `mod`(a[i]^e,m);" "6#/&%\"bG6#%\"iG-%$modG6$)&%
\"aG6#F'%\"eG%\"mG" }{TEXT -1 41 " and to send her the sequence of num
bers " }{XPPEDIT 18 0 "b[1];" "6#&%\"bG6#\"\"\"" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "b[2];" "6#&%\"bG6#\"\"#" }{TEXT -1 9 ", . . ., " }
{XPPEDIT 18 0 "b[k];" "6#&%\"bG6#%\"kG" }{TEXT -1 23 ". He need not ke
ep the " }{XPPEDIT 18 0 "b[i];" "6#&%\"bG6#%\"iG" }{TEXT -1 64 " 's se
cret after he has  computed them. But, he should keep the " }{XPPEDIT 
18 0 "a[i];" "6#&%\"aG6#%\"iG" }{TEXT -1 10 "'s secret." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "9. Now when Alice ge
ts the " }{XPPEDIT 18 0 "b[i];" "6#&%\"bG6#%\"iG" }{TEXT -1 28 " 's fr
om Bob, she computes  " }{XPPEDIT 18 0 "`mod`(b[i]^d,m);" "6#-%$modG6$
)&%\"bG6#%\"iG%\"dG%\"mG" }{TEXT -1 34 " and this will turn out to be \+
the " }{XPPEDIT 18 0 "a[i];" "6#&%\"aG6#%\"iG" }{TEXT -1 96 "'s. Then \+
she can use Maple to convert these back into text as we illustrate in \+
the next section." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 30 "Binary Exponentiation Modulo m" }}{PARA 
0 "" 0 "" {TEXT -1 70 "A key component in implementing the RSA scheme \+
is the computation of  " }{XPPEDIT 18 0 "`mod`(x^e,m);" "6#-%$modG6$)%
\"xG%\"eG%\"mG" }{TEXT -1 111 " for large positive integers x, e, and \+
 m.  An obvious, but infeasible, way to do this is to try the command \+
:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x^e mod m;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 319 "The problem with this approach is
 that x^e is computed first. If x and e are large,  x^e may be larger \+
than the number of particles in the universe.  So storing this number \+
to divide it by m is not possible. For large enough integers x,  m and
 e this may even cause Maple to crash. Instead,  one should use the co
mmand " }{TEXT 354 16 "Power(x,e) mod m" }{TEXT -1 49 ".  This procedu
re implements an algorithm called " }{TEXT 355 30 "binary exponentiati
on modulo m" }{TEXT -1 444 ". The basic idea is to  perform a series o
f operations of the form y:=x^2 mod m.  This will give powers of the f
orm x^N mod m where N is a power of 2. Some of these can be multiplied
 together to obtain x^e mod m. The algorithm is actually pretty simple
, but amazingly efficient. One can see the Maple code for Power(x,e) m
od m (which is the same as `mod/Power`(x,e,m) ) as follows.  Aside fro
m the error checking the procedure is quite simple. " }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "interface
(verboseproc=3);\nprint(`mod/Power`);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 39 "\n\nLet us compare the time required for " }{TEXT 356 9 "
2^e mod m" }{TEXT -1 5 " and " }{TEXT 357 16 "Power(2,e) mod m" }
{TEXT -1 9 " for e = " }{XPPEDIT 18 0 "10^6;" "6#*$\"#5\"\"'" }{TEXT 
-1 129 " and m = 10^100+3.  If we tried to increase the value of e her
e we would quickly find that the first method would cause trouble.\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "m:=10^100+3;\ne:=10^6;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "st:=time():\na:=2^e mod m:
\nelapsed_time:=(time()- st)*sec;\na;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "st:=time():\nb:=Power(2,e) mod m:\nelapsed_time:=(tim
e()- st)*sec;\nb;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 155 "\nUsing t
he first method for large e will be unfeasible, however, note how rapi
dly the binary exponentiation method works for the following large int
egers:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "die:=rand(1..10
^100):\nx:=die();\ne:=die();\nm:=die();" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "st:=time():\na:=Power(x,e) mod m:\nelapsed_time:=(tim
e() - st)*sec;\na;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "An Example of the RSA System" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "We illustrate the RSA system using
 two randomly chosen primes with 20 decimal digits.  Note that " }
{XPPEDIT 18 0 "10^(n-1);" "6#)\"#5,&%\"nG\"\"\"F'!\"\"" }{TEXT -1 46 "
  is the smallest n decimal digit integer and " }{XPPEDIT 18 0 "10^n-1
;" "6#,&)\"#5%\"nG\"\"\"F'!\"\"" }{TEXT -1 93 " is the largest.  For e
xample if n = 3, we obtain the smallest and largest 3 digit integers:
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "10^2; 10^3-1;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We first generate two 20 digit ran
dom positive integers and then use the Maple command " }{TEXT 332 9 "n
extprime" }{TEXT -1 110 " to obtain the smallest primes greater than e
ach number. Notice that in rand we use the range 10^19..10^20-1.\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "N:=rand(10^19..10^20 -1)();
\nM:=rand(10^19..10^20 -1)();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "
Now we use the command nextprime. The command " }{TEXT 358 12 "nextpri
me(n)" }{TEXT -1 41 " finds the smallest prime greater than n." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p:=nextprime(N);\nq:=nextpri
me(M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Now we multiply these t
ogether to form our " }{TEXT 333 7 "modulus" }{TEXT -1 3 " m:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m:=p*q;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 43 "How many digits does m have? This tells us:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "length(m);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 15 "Now we compute " }{XPPEDIT 18 0 "phi(m);" "6#-%
$phiG6#%\"mG" }{TEXT -1 98 " which is possible since we know that m = \+
p*q where p and q are primes. We simply use the formula " }{XPPEDIT 
18 0 "phi(m) = (p-1)*(q-1);" "6#/-%$phiG6#%\"mG*&,&%\"pG\"\"\"F+!\"\"F
+,&%\"qGF+F+F,F+" }{TEXT -1 30 ". [Note that this formula for " }
{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG" }{TEXT -1 156 " works only \+
when m is a product of two different primes. If you study number theor
y,  you may learn more about this function.] We denote it by the varia
ble " }{TEXT 359 4 "phi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 
"phi:=(p-1)*(q-1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now we wish
 to find a number e such that " }{XPPEDIT 18 0 "igcd(e,phi(m)) = 1;" "
6#/-%%igcdG6$%\"eG-%$phiG6#%\"mG\"\"\"" }{TEXT -1 71 ".  We keep tryin
g random 20 digit integers till we find one that works." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for i from 1 do\ne:=rand(10^19..10^
20-1)();\nif igcd(e,phi) = 1 then break; fi;\nod:\ne;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 360 23 "inverse of e modulo p
hi" }{TEXT -1 27 " is an integer d such that " }{TEXT 361 15 "e*d mod \+
phi = 1" }{TEXT -1 168 ". From number theory it is known that an integ
er e has an inverse mod phi if and only igcd(e,phi) = 1.  The followin
g command will compute the inverse of e modulo phi:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "d:=1/e mod phi;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 23 "\nWe check that e*d mod " }{XPPEDIT 18 0 "phi(m)
" "6#-%$phiG6#%\"mG" }{TEXT -1 6 " = 1.\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "e*d mod phi;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 
"So the " }{TEXT 263 10 "public key" }{TEXT -1 41 " Alice sends to Bob
 and mades public is \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
Public_Key:=[m,e];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "\nThe infor
mation we must keep secret in this case is:\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "Private_key:=[p,q,d,phi];" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 18 "\nIn cryptography, " }{TEXT 265 10 "plain text" }
{TEXT -1 77 " refers to any message that is not enciphered. After encr
yption it is called " }{TEXT 266 11 "cipher text" }{TEXT 362 2 ". " }
{TEXT -1 46 "We assign Bob's message to the variable named " }{TEXT 
289 10 "plain_text" }{TEXT -1 70 ", as follows. (Of course, the name o
f the variable is not important.)\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 275 "plain_text:=\"It is not knowledge, but the act of le
arning, not possession but the act of getting there, which grants the \+
greatest enjoyment. When I have clarified and exhausted a subject, the
n I turn away from it, in order to go into darkness again; --Karl Frie
drich Gauss\";\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Note that th
e text includes new line symbols, \\n, as well as numerous blanks.The \+
\\ at the end of each line is just Maple's way of saying that this sho
uld all be on one line, but I am forced to break it at this point." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "\nWe now convert this text to a l
ist of numbers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "List1:=c
onvert(plain_text,bytes);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "We \+
consider List1 as the base 256 representation of a number.  We convert
 this to a base m number as follows:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "List2:=convert(List1,base, 256,m);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 37 "Now we apply the mapping which sends " }{TEXT 
363 1 "x" }{TEXT -1 4 " to " }{TEXT 364 16 "Power(x,e) mod m" }{TEXT 
-1 20 "  to each number in " }{TEXT 334 5 "List2" }{TEXT -1 25 ". This
 can be done using " }{TEXT 264 5 "map. " }{TEXT -1 67 " We thus obtai
n the enciphered message that will be sent to Alice. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "cipher_t
ext:=map(x->Power(x,e) mod m, List2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 116 "\nNow Bob sends this to Alice and the whole world for th
at matter. But since only Alice has the secret  private key, " }{TEXT 
290 45 "only she will be able to decipher the message" }{TEXT -1 61 " \+
(in the case where p and q are chosen with sufficient care.)" }}{PARA 
0 "" 0 "" {TEXT -1 80 "\nWe show now how she deciphers the message: Sh
e applies the mapping which sends " }{TEXT 365 1 "x" }{TEXT -1 4 " to \+
" }{TEXT 366 16 "Power(x,d) mod m" }{TEXT -1 7 " where " }{TEXT 367 1 
"d" }{TEXT -1 44 " is her private key to each received number:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "decipher1:=map(x->Power(x,d)
 mod m,cipher_text);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Next she \+
converts this to base 256:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "decipher2:=convert(decipher1,base,m,256);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 81 "Finally she converts the numbers back to text and sees
 what message Bob sent her:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "decipher3:=convert(decipher2,bytes);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Breaking th
e RSA System" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "To decipher the me
ssage it is " }{TEXT 335 8 "believed" }{TEXT -1 112 " (but not proved)
 that one must be able to find the private key d given only the public
 key [m,e]. If one knows " }{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG
" }{TEXT -1 54 " then Maple quickly computes d by the command 1/e mod \+
" }{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG" }{TEXT -1 30 ".  It is k
nown that computing " }{XPPEDIT 18 0 "phi(m);" "6#-%$phiG6#%\"mG" }
{TEXT -1 733 " is the same order of difficulty as factoring m. If m is
 the product of two 100 digit primes, say, then m is 200 digits long a
nd current factoring algorithms are unable to factor such numbers with
in a reasonable time unless the primes are poorly chosen (as in Proble
m 4 in this week's homework assignment).  Don't ask Maple to factor su
ch a number for you unless you want to wait a long time -- years or ce
nturies, depending on the size of the number and luck. Factoring algor
ithms always depend on luck. After all one might guess the factors, bu
t this would be very unlikely. Of course, the technology is changing r
apidly. Mathematicians are finding better algorithms and Engineers are
 building faster computers every day it seems. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "It is also " }{TEXT 269 
8 "believed" }{TEXT -1 184 " (but not proved) that breaking the RSA sy
stem is equivalent to factoring the modulus m. The best algorithm know
n at present for factoring such large, difficult numbers, is called th
e " }{TEXT 270 38 "Number Field Sieve Factoring Algorithm" }{TEXT -1 
35 ". A group of mathematicians called " }{TEXT 267 9 "The Cabal" }
{TEXT -1 79 " announced the completion, on November 14, 2000, of the f
actorization with the " }{TEXT 268 33 "Special Number Field Sieve (SNF
S)" }{TEXT -1 36 " of the 233-digit Cunningham number " }{XPPEDIT 18 
0 "2^773+1;" "6#,&*$\"\"#\"$t(\"\"\"F'F'" }{TEXT -1 748 " into the pro
duct of the numbers 3 and 533371 and three primes of  55, 71, and 102 \+
digits, respectively. This established a new record for the Special Nu
mber Field Sieve.  [Note that this is somewhat easier to factor than a
 number of this size which is the product of only two primes of approx
imate equal size.  In this case the 3 and 533371 factors were easy to \+
find, and the remaining factor has the \"small\" 55-digit and 71-digit
 primes.] \n\nThe sieving was done on about 150 SGI workstations and S
un workstations and servers running at 180-450 MHz, and on about 100 P
Cs running at 266-600 MHz. The sieving took about five calendar months
. It was started mid-April, 2000, and finished on September 15, 2000. \+
Total sieving time was 57.4 CPU years. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 67 "For more information on factoring an
d the RSA scheme take a look at" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "    " }{TEXT 368 64 "http://www.rsasecurit
y.com/rsalabs/challenges/factoring/faq.html" }}{PARA 0 "" 0 "" {TEXT 
-1 29 "\nHere you can read about the " }{TEXT 369 23 "RSA Factoring Ch
allenge" }{TEXT -1 660 ", an effort, sponsored by RSA Laboratories, to
 encourage research into the factoring of large numbers of the type us
ed in RSA keys. A set of eight challenge numbers, ranging in size from
 576 bits to 2048 bits is posted here. Each number is the product of t
wo large primes, similar to the modulus of an RSA key pair. Prize awar
ds from $10,000 to $200,000 are offered for the prime factors of these
 eight numbers.\n\nThere are many sources of information on the RSA sy
stem. It is important not only for sending data securely but also can \+
be used to \"sign\" electronically submitted documents. For more infor
mation see for example any of the many books available on " }{TEXT 
271 14 "cryptography. " }{TEXT -1 81 " Just search on this key word us
ing Web Luis or on your favorite search engine.  " }{TEXT 272 21 "RSA \+
Data Security Inc" }{TEXT -1 51 ". now controls the RSA system. The co
mpany's URL is" }{TEXT 273 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT 370 36 "       http://www.rsasecurity.com/. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Both " }
{TEXT 371 8 "Netscape" }{TEXT -1 5 " and " }{TEXT 372 17 "Internet Exp
lorer" }{TEXT -1 64 " contain security software from RSA Data Security
 Inc. Also the " }{TEXT 337 3 "PGP" }{TEXT -1 226 " (Pretty Good Priva
cy) encryption system incorporates the RSA scheme.  PGP can be used fo
r authentication purposes (i.e., to electronically sign a document). Y
ou may download the program for free from the MIT website\n\n        \+
" }{TEXT 373 35 "http://web.mit.edu/network/pgp.html" }{TEXT 336 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "You ma
y find out about Philip Zimmermann, the creator of PGP, and his troubl
es with the government at his webpage\n\n        " }{TEXT 374 23 "http
://web.mit.edu/prz/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 124 "He even has his phone number given on the webpage, but
 in a way that forces people to read an entire paragraph to obtain it.
" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " readdata, \+
writeto and appendto" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 553 "If one wants to run a Maple program that will take sev
eral days to complete the computation and that requires that data be p
rinted out along the way, it is wise to print this to a file. This way
 if Maple crashes before it finished, at least some data can be obtain
ed from the file. Also there are times when you might obtain data from
 other sources that you wish Maple to process. In this case if the dat
a is in a text file, it may be read into a Maple worksheet so that you
 can process it further. Here are some commands that will allow one to
 do this." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "writeto and appendto
" }}{EXCHG {PARA 15 "" 0 "" {TEXT -1 17 "When the command " }{TEXT 
291 18 "writeto(filename);" }{TEXT -1 79 " is executed all future comm
ands will have their results immediately stored in " }{TEXT 338 8 "fil
ename" }{TEXT -1 129 " and will not be displayed on the screen. Unless
 you indicate otherwise the file will be created and stored in your Ma
ple folder." }}{PARA 15 "" 0 "" {TEXT -1 115 "While in this mode, no p
rompt will appear at the terminal. The prompt and the input statements
 will be echoed into " }{TEXT 339 8 "filename" }{TEXT -1 33 " (unless \+
echoing is turned off). " }}{PARA 15 "" 0 "" {TEXT -1 20 "The special \+
command " }{TEXT 292 17 "writeto(terminal)" }{TEXT -1 73 " will restor
e the standard mode of printing the results on the terminal. " }}
{PARA 15 "" 0 "" {TEXT -1 3 "If " }{TEXT 340 8 "filename" }{TEXT -1 
58 " already exists the contents of the file are overwritten. " }}
{PARA 15 "" 0 "" {TEXT -1 62 "An alternative mode of writing to a file
 is available via the " }{TEXT 294 8 "appendto" }{TEXT -1 26 " functio
n. In the case of " }{TEXT 293 19 "appendto(filename);" }{TEXT -1 50 "
 rather than overwriting the contents (if any) of " }{TEXT 341 8 "file
name" }{TEXT -1 20 ", the new output is " }{TEXT 343 8 "appended" }
{TEXT -1 4 " to " }{TEXT 342 8 "filename" }{TEXT -1 21 ". In other res
pects, " }{TEXT 295 8 "appendto" }{TEXT -1 5 " and " }{TEXT 296 7 "wri
teto" }{TEXT -1 30 " have the same functionality. " }}{PARA 0 "" 0 "" 
{TEXT -1 47 "To see what's going on we will need to use the " }{TEXT 
344 8 "readdata" }{TEXT -1 9 " function" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
119 "writeto(`datafile1`);\nfor i from 10^10 to 10^10 + 6 do\nprintf(
\"%15d %15d %15d \\n\",i, 5*i, 10*i);\nod;\nwriteto(terminal);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 8 "readdata" }}{PARA 15 "" 0 "" {TEXT -1 4 "The " }{TEXT 
297 9 "readdata " }{TEXT -1 341 "function reads numeric data from an i
nput text file into Maple. The data must consist of integers or floats
 arranged in columns separated by white space. If only one column of d
ata is read, the output is a list of the data. If more than one column
 is read, the output is a list of rows of data corresponding to the ro
ws of data in the file. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
L:=readdata(`datafile1`, integer, 3);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "Here are two ways to display the date read in." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "matrix(L);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 71 "for i from 1 to nops(L) do \nprintf(\" %1
5d %15d %15d \\n\",op(L[i])); \nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Other read
 and write commands" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 175 "Maple has a variety of commands for reading data
 from external files and writing data to external files. The type of c
ommand you use depends on the type of data. For example, " }{TEXT 308 
8 "readdata" }{TEXT -1 51 " will work for lists of numbers, but you wi
ll need " }{TEXT 309 8 "readline" }{TEXT -1 15 " to read text (" }
{TEXT 310 7 "strings" }{TEXT -1 168 "). Similarly for writing to a fil
e, the type of data is important. Here is a brief summary of the other
 commands. For more details see the online help for each command." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 10 "For inpu
t:" }{TEXT -1 5 "\n\n1. " }{TEXT 298 8 "readline" }{TEXT -1 65 " reads
 a single line from a file or the terminal, and returns it " }{TEXT 
314 11 "as a string" }{TEXT -1 94 ".  You can then parse this with \"s
scanf\" or the \"parse\" function (for Maple expressions).\n\n2. " }
{TEXT 299 8 "readdata" }{TEXT -1 19 " reads a text file " }{TEXT 313 
18 "containing numbers" }{TEXT -1 149 " separated by tabs or spaces, a
nd returns it as a list.  You can specify which columns to read, and w
hether to treat them as integers or floats.\n\n3. " }{TEXT 300 26 "fsc
anf , sscanf, and scanf" }{TEXT -1 150 " are based on the C standard l
ibrary functions of the same names for formatting input.  The Maple ve
rsions of these functions each return a list.\n\n4. " }{TEXT 301 9 "re
adbytes" }{TEXT -1 29 " reads a specified number of " }{TEXT 315 5 "by
tes" }{TEXT -1 80 " from a file, and returns either a list of integers
 or a string of characters.\n\n" }{TEXT 312 11 "For output:" }{TEXT 
-1 5 "\n\n1. " }{TEXT 302 9 "writeline" }{TEXT -1 8 " writes " }{TEXT 
316 8 "strings " }{TEXT -1 31 "to a file or the terminal.\n\n2. " }
{TEXT 303 9 "writedata" }{TEXT -1 18 " is used to write " }{TEXT 317 
14 "numerical data" }{TEXT -1 21 " to a text file.\n\n3. " }{TEXT 304 
29 "fprintf , sprintf, and printf" }{TEXT -1 198 " are based on the C \+
standard library functions of the same names; these allow you to contr
ol the format of output.  There is also an \"nprintf\" which returns a
 Maple symbol rather than a string.\n\n4. " }{TEXT 305 10 "writebytes
" }{TEXT -1 12 " writes the " }{TEXT 318 5 "bytes" }{TEXT -1 120 " fro
m a string of characters or list of integers to a file (the file is op
ened with type text or binary, respectively).\n" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "5. " }{TEXT 306 7 "writeto" }{TEXT -1 4 " or " }{TEXT 307 
8 "appendto" }{TEXT -1 8 " writes " }{TEXT 319 22 "output from the scr
een" }{TEXT -1 41 " into the specified file. In the case of " }{TEXT 
320 8 "writeto," }{TEXT -1 45 " the old file is overwritten, in the ca
se of " }{TEXT 321 9 "appendto," }{TEXT -1 65 " the new data produced \+
by Maple is appended to the existing file." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 41 "Converting to Fortran, C, Latex, and HTML
" }}{PARA 0 "" 0 "" {TEXT -1 5 "\nThe " }{TEXT 35 7 "fortran" }{TEXT 
-1 69 " function generates Fortran code for evaluating the input. The \+
input " }{TEXT 35 1 "s" }{TEXT -1 107 " must be a single algebraic exp
ression, an array of algebraic expressions, a list of equations of the
 form " }{TEXT 35 16 "name = algebraic" }{TEXT -1 86 " that is underst
ood to mean a sequence of assignment statements, or a Maple procedure.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(codegen,fortran):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f:=unapply(sin(x*y) + ln(
x^2 + 5*x*y + 1), x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "
fortran(f);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 35 1 "
C" }{TEXT -1 63 " function generates C code for evaluating the input. \+
The input " }{TEXT 35 1 "s" }{TEXT -1 129 " must be one of the followi
ng: a single algebraic expression, an array of algebraic expressions, \+
a list of equations of the form " }{TEXT 35 16 "name = algebraic" }
{TEXT -1 89 " (which is understood to mean a sequence of assignment st
atements), or a Maple procedure." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "with(codegen,C):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "C(f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n
" }{TEXT -1 4 "The " }{TEXT 322 5 "latex" }{TEXT -1 85 " function prod
uces output which is suitable for printing the input expression with a
 " }{TEXT 323 8 "LaTeX 2e" }{TEXT -1 82 " processor. It knows how to f
ormat integrals, limits, sums, products and matrices." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=matrix([[seq(i,i=1..5)],[seq(i^2
,i=1..5)]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(A);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 324 6 "export" }
{TEXT -1 19 " a worksheet as an " }{TEXT 325 4 "HTML" }{TEXT -1 40 " d
ocument for viewing in a Web browser: " }}{PARA 14 "" 0 "" {TEXT -1 
48 " 1. Open the worksheet that you want to export. " }}{PARA 14 "" 0 
"" {TEXT -1 13 " 2. From the " }{TEXT 36 4 "File" }{TEXT -1 14 " menu,
 choose " }{TEXT 36 9 "Export As" }{TEXT -1 11 ", and then " }{TEXT 
36 4 "HTML" }{TEXT -1 149 ". \n\nNote that when Maple creates the HTML
 file it will also create a folder containing gif files that will be e
mployed when the HTML file is opened. " }}{PARA 14 "" 0 "" {TEXT -1 0 
"" }}{PARA 14 "" 0 "" {TEXT -1 693 "You may use the above to convert t
he entire worksheet to an HTML file.  Or, individual plots can be save
d in a separate worksheet and then exported to html. This will create \+
a file containing only the plot. The animated plots will run continuou
sly when opened with a web browser. \n\nTo see this execute the the fo
llowing animation. Then select the output of the animation. Open a new
 worksheet and paste it into the new worksheet. Then follow the above \+
instructions to export as an html document. Note that this will also c
reate a gif file which will be placed in a some folder. After you have
 done this open the html file with Netscape or Internet Explored and y
ou will  see a \"flying plane\". " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}
{PARA 14 "" 0 "" {TEXT -1 44 "If you want to experiment you may go to \+
the " }{TEXT 376 4 "File" }{TEXT -1 19 " menu item, select " }{TEXT 
375 7 "Save As" }{TEXT -1 51 ", and when the window comes up choose fo
r filetype " }{TEXT 377 11 "HTML Source" }{TEXT -1 73 ". This will sav
e the entire worksheet. You can open it with a webbrowser." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "plo
ts[animate3d]( 10*sin(t)*abs(y)*abs(x), x = -1..1,y=-3..3, t=0..2*Pi, \+
style=patchnogrid);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Writing Efficient Maple Code" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 215 "For a nu
mber of useful hints on how to write efficient code, that is, how to i
mprove the performance of a program that you have working, but would l
ike to speed up, read the help page that comes up when you execute " }
{TEXT 330 10 "?efficient" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?efficient" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 36 "Assignment 12 -- Wednesday, April 24" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 274 10 "Problem 1." }
{TEXT -1 98 "  Alice sent a message to Bob using the RSA system. The p
ublic key used to encode the message was " }}{PARA 0 "" 0 "" {TEXT -1 
79 "[m,e] = [187454749788911503119994043213682616233000961, 2307369747
4256143563]. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "The enciphered message was" }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 1228 " [9976308850671705071637949852035
7841718875740, 120165413768425112311337085191282286020515442, 13055485
2152844850464253187688342192964186591, 3083103880347330918694403904469
3095063005267, 43791870977158745785820581340119798552660187, 699699650
42359538654291028155927734707714502, 338699659256999909038017914779294
60546990002, 33844215332363231914638130818590411552396086, 61034870951
899306892357141802983347226683758, 15268070224895082795829529457271713
5277913525, 109591116592896191360922558098912453060002989, 15548994778
9995346755966254731407091292570943, 3050853785484135288833738680013167
8916944679, 45657204694505066046067342403378233363404150, 122580283375
455701674982424288406984864898071, 63190645718352016104824679398188140
019415424, 17253386795565064409576412640690151622064852, 7643268136560
2448262013510061731737725777808, 4419855123469897669096067167819380964
828479, 87913005918401414546998775379426321338719933, 1554345165267352
15629181544556450169276890711, 746953813599011144431688669057557230392
88800, 246854149837977729662442017899408430336317, 1217662277746125438
67361050340041336298158533, 166784352080096633524033563364825999607054
575, 72405993088552642775735865911360107823805504]\n\nDecipher this me
ssage. " }{TEXT 345 4 "Hint" }{TEXT -1 53 ". Use of ifactor with the o
ption 'pollard' may help. " }{TEXT 346 112 "When you are finished with
 the decription. You should use colons where possible to suppress unne
cessary output. " }{TEXT -1 2 "\n\n" }{TEXT 275 10 "Problem 2." }
{TEXT -1 176 "  Find, using Maple, a triple [m,e,d] where m is the pro
duct of two 200 decimal digit primes, e is a random prime number of 10
0 decimal digits and d is the inverse of e modulo " }{XPPEDIT 18 0 "ph
i(m);" "6#-%$phiG6#%\"mG" }{TEXT -1 168 ". Use this to encipher some m
essage of your choice of at least 4 lines length using the RSA system.
 Then, decipher the enciphered text to obtain the original message.\n
\n" }{TEXT 276 9 "Problem 3" }{TEXT -1 55 ". Given that  m = pq where \+
p and q are primes and that " }{XPPEDIT 18 0 "phi(m) = (p-1)*(q-1);" "
6#/-%$phiG6#%\"mG*&,&%\"pG\"\"\"F+!\"\"F+,&%\"qGF+F+F,F+" }{TEXT -1 
15 ". Suppose that " }{XPPEDIT 18 0 "m = 10^100+598*10^50+67497;" "6#/
%\"mG,(*$\"#5\"$+\"\"\"\"*&\"$)fF)*$F'\"#]F)F)\"&(\\nF)" }{TEXT -1 6 "
 and  " }{XPPEDIT 18 0 "phi(m) = 10^100+596*10^50+66900;" "6#/-%$phiG6
#%\"mG,(*$\"#5\"$+\"\"\"\"*&\"$'fF,*$F*\"#]F,F,\"&+p'F," }{TEXT -1 69 
". Find p and q.  Check that the factors you find are indeed primes.\n
\n" }{TEXT 327 6 "Hint: " }{TEXT -1 41 "You have two equations and two
 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 277 9 "Problem 4
" }{TEXT -1 232 ". [This exercise shows that if care is not taken in c
hoosing the primes p and q used in the RSA system that the system can \+
be easily broken.]  Suppose you know that a person's public key modulu
s m is the following 200 digit number :" }}{PARA 0 "" 0 "" {TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "1
6657182008590549877293859320735990648731577415651414508956413467009247
6753082292235735087737299144087247477492337792847654146253083529676393
00872282279199825030962210816740376886176930271167988018683;" "6#\"cw$
o=!))z;r-$p<')oPSn\"3@i4.D)*>zAGs3IRw'HN3`i9aw%GzPB\\xuC(3W\"*HPx3NdB#
H#3`nZ#4qY8k&*3XT^cTx:t[1*ft?$fQHx)\\0f3?=dm\"" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 326 65 "Factor m and check your factorization after th
e factors are found" }{TEXT -1 3 ".  " }{TEXT 281 18 "Use the followin
g " }{TEXT 278 27 "Fermat factorization method" }{TEXT 282 2 ": " }
{TEXT -1 15 " Note that if  " }{XPPEDIT 18 0 "m = x^2-y^2;" "6#/%\"mG,
&*$%\"xG\"\"#\"\"\"*$%\"yGF(!\"\"" }{TEXT -1 6 " then " }{XPPEDIT 18 
0 "m = (x-y)*(x+y);" "6#/%\"mG*&,&%\"xG\"\"\"%\"yG!\"\"F(,&F'F(F)F(F(
" }{TEXT -1 8 ". So if " }{XPPEDIT 18 0 "1 < x-y;" "6#2\"\"\",&%\"xGF$
%\"yG!\"\"" }{TEXT -1 198 " we have a non-trivial factorization of m. \+
If m is a product of just two primes then one will be x - y and the ot
her will be x + y.  This is the basis of the Fermat factorization meth
od. Note that  " }{XPPEDIT 18 0 "m = x^2-y^2" "6#/%\"mG,&*$%\"xG\"\"#
\"\"\"*$%\"yGF(!\"\"" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 
0 "x^2-m = y^2;" "6#/,&*$%\"xG\"\"#\"\"\"%\"mG!\"\"*$%\"yGF'" }{TEXT 
-1 35 ".  So if for some number x we have " }{TEXT 279 34 "type(sqrt(x
^2 - m),integer) = true" }{TEXT -1 17 " then we can set " }{TEXT 347 
18 "y = sqrt(x^2 - m) " }{TEXT -1 70 "and using the above idea we may \+
factor m.  Fermat's method is to take " }{TEXT 348 4 "x = " }{XPPEDIT 
349 0 "ceil(sqrt(m));" "6#-%%ceilG6#-%%sqrtG6#%\"mG" }{TEXT -1 59 " an
d keep incrementing x by 1 till an x is found such that " }{XPPEDIT 
350 0 "x^2-m;" "6#,&*$%\"xG\"\"#\"\"\"%\"mG!\"\"" }{TEXT -1 12 " is a \+
square" }{TEXT 280 3 ".  " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "[Note that you may need to use " }
{TEXT 283 24 "ceil(evalf(sqrt(m),200))" }{TEXT -1 3 ".] " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 272 "If the number m i
s a product of two primes that are relatively close together then this
 method will factor m quickly. If you do not find the factors in just \+
a few seconds, you probably have a bug in your program. For some reaso
n Maple does not have this method built in. \n" }}}}}{MARK "3" 0 }
{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 1 33 1 1 }