Homework 11 Solution
David Stachnik
The RSA encryption algorithm is a method of securely transmitting information without needing to
exchange secret encryption keys. Instead, separate encryption and decryption keys are used in a way where
only the decryption key (private key) must remain secret and is not exchanged between the parties. The
encryption key (public key) may then be published so everyone has access to it. Each party has a public key
and private key which are unique to that party.
Consider the two classic cryptographic parties Alice and Bob. If Alice wants to send a secure message
to Bob, she encrypts the message using Bob’s public key. Now only Bob can decrypt the message using his
private key (which only he has). Additionally, Alice can sign the message (so Bob knows it’s from Alice)
by encrypting some ’signature’ text using her private key before encrypting the entire message (including
signature) using Bob’s public key. When Bob decrypts the message, only the signature is still garbled. Then
Bob decrypts the signature using Alice’s public key to ensure that Alice was the sender. This is known as a
’oneway’ or ’trap door’ cipher, of which RSA encryption is an example.
The actual choice of the public and private keys is the RSA algorithm. Essentially, it relies on some
properties of modular algebra (where the result is taken modulo
n
after each operation, for an integer
n
)
which show that (
M
e
)
d
=
M
(mod
n
) for a suitable choice of
e
,
d
, and
n
.
Thus a message
M
can be
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 Spring '08
 BOZAROV
 Cryptography, Prime number, decryption key, RSA algorithm

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