MATH 135
Fall 2007
Assignment #8
Due: Thursday 22 November 2007, 8:20 a.m.
Notes
:
•
In problems 3 and 4, please do your work by hand and show this work.
(Of course, using
MAPLE to check your answers is probably a good idea.)
•
The last page of this document contains some handy MAPLE commands.
•
The text file “Problem 6 Values” contains the values of the large numbers in Problem 6 in text
format.
•
There will tentatively be a TA in the Tutorial Centre to help with Maple issues from 1:30 to
2:30 on Wednesday 21 November.
HandIn Problems
1. Determine the complete solution to the congruence
x
31
+ 17
x
≡
43 (mod 55).
2. Prove that
n
21
≡
n
3
(mod 21) for every integer
n
.
3. If
p
= 251,
q
= 281 and
e
= 73, find the associated RSA public and private keys.
4.
(a) In
an
RSA
scheme,
the
public
key
(
e, n
)
=
(23
,
19837)
and
the
private
key
is
(
d, n
) = (17819
,
19837). Using the Square and Multiply algorithm,
encrypt
the message
M
= 1975 using the appropriate key.
(b) In an RSA scheme,
the the public key is (
e, n
)
=
(801
,
1189),
the private key is
(
d, n
) = (481
,
1189).
Note that 41 is a factor of 1189.
Using the Chinese Remainder
Theorem,
decrypt
the ciphertext
C
= 567 using the appropriate key.
5. Complete the following steps using MAPLE:
•
Using the command
restart
, clear MAPLE’s memory.
•
Using the command
rand
, create two random integers
R1
and
R2
each less than 10
50
.
•
Using the command
nextprime
, find the next prime larger than each of
R1
and
R2
. (Call
these
p
and
q
. Use the command
isprime
to verify that each is prime.)
•
Calculate
n
and
φ
(
n
), calling them
n
and
phi
.
•
Define
e
to be your UWID number. Using the command
gcd
, check if the gcd of
e
and
phi
is 1. If it is, go to the next step. If it is not, repeat this process to find the first value
of
e
larger than your UWID that gives a gcd of 1 with
phi
.
•
Using the command
msolve
, determine the value of
d
, part of the private key. (After using
msolve
, you will have to define
d
by copying and pasting your result.)
•
Define
M
to be
2625242322212019181716151401020304050607080910111213
.
•
Using the
&^
command, encrypt
M
using
e
and
n
, to get
C
.
•
Decrypt
C
using
d
and
n
.
Print your output and hand it in with your Assignment.
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6. Complete the following steps using MAPLE:
•
Using the command
restart
, clear MAPLE’s memory.
•
Define
n
to be
28389368948326183561923542922514268376927952930167092496012482257483771333584
99454468046751209461074038548632181358620800764412077
(You may find it helpful to copy and paste the value of
n
available in the accompanying
text file instead of trying to (correctly!) retype
n
.)
•
Define
e
to be
123578176234876198276491234876148973
and
d
to be
14120514684168224392251360605222273900386543069562554506259906955949044365472
03725494350063143836462319733064004787101733886580301
•
Suppose your public key is (
e, n
) and your private key is (
d, n
) and that you receive the
message
C
, which is
80128892902547886808096146110599124584248309969798722917949036154729920471947
1173555177025839977041228416176205654413823029580356
Decrypt the message using the appropriate key.
•
Using the correspondence
A
↔
01
,
B
↔
02
, etc., write the message in English.
(Note the order of letters in the English alphabet is
ABCDEFGHIJKLMNOPQRSTUVWXYZ
. You
will probably find it easiest to do this last step by hand.)
Print your output and hand it in with your Assignment.
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 Fall '08
 ANDREWCHILDS
 Math

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