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Massachusetts Institute of Technology
6.042J/18.062J, Spring ’10
: Mathematics for Computer Science
April 2
Prof. Albert R. Meyer
revised March 29, 2010, 739 minutes
Solutions to InClass Problems Week 8, Fri.
Problem 1.
Let’s try out RSA! There is a complete description of the algorithm at the bottom of the page. You’ll
probably need extra paper.
Check your work carefully!
(a)
As a team, go through the
beforehand
steps.
• Choose primes
p
and
q
to be relatively small, say in the range 1040. In practice,
p
and
q
might
contain several hundred digits, but small numbers are easier to handle with pencil and paper.
• Try
e
= 3
,
5
,
7
,...
until you ﬁnd something that works. Use Euclid’s algorithm to compute
the gcd.
• Find
d
(using the Pulverizer —see appendix for a reminder on how the Pulverizer works —or
Euler’s Theorem).
When you’re done, put your public key on the board. This lets another team send you a message.
(b)
Now send an encrypted message to another team using their public key. Select your message
m
from the codebook below:
• 2 = Greetings and salutations!
• 3 = Yo, wassup?
• 4 = You guys are slow!
• 5 = All your base are belong to us.
• 6 = Someone on
our
team thinks someone on
your
team is kinda cute.
• 7 = You
are
the weakest link. Goodbye.
(c)
Decrypt the message sent to you and verify that you received what the other team sent!
RSA Public Key Encryption
Creative Commons
2010,
Prof. Albert R. Meyer
.
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Solutions
to
InClass
Problems
Week
8,
Fri.
Beforehand
The receiver creates a public key and a secret key as follows.
1. Generate two distinct primes,
p
and
q
.
2. Let
n
=
pq
.
3. Select an integer
e
such that
gcd(
e,
(
p
−
1)(
q
−
1))
=
1
.
The
public
key
is the pair
(
e,n
)
. This should be distributed widely.
4. Compute
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 Spring '11
 Prof.AlbertR.Meyer
 Computer Science

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