CS 70
Discrete Mathematics and Probability Theory
Fall 2010
Tse/Wagner
Note 6
PublicKey Cryptography, RSA, and Modular Arithmetic
This lecture, we’ll discuss a beautiful and important application of modular arithmetic: the
RSA publickey
cryptosystem
, so named after its inventors Ronald Rivest, Adi Shamir, and Leonard Adleman.
The basic setting for cryptography is typically described via a cast of three characters: Alice and Bob, who
wish to communicate conﬁdentially over some (insecure) communication link, and Eve, an eavesdropper
who is listening in and trying to discover what they are saying. Suppose Alice wants to transmit a message
x
(written in binary) to Bob. Alice will apply her encryption function
E
to
x
and send the encrypted message
E
(
x
)
over the communication link. Bob, upon receipt of
E
(
x
)
, will then apply his decryption function
D
to
it and thus recover the original message: i.e.,
D
(
E
(
x
)) =
x
.
Since the link is insecure, Alice and Bob have to assume that Eve may get hold of
E
(
x
)
. (Think of Eve
as being a “sniffer” on the network.) Thus ideally we would like to know that the encryption function
E
is
chosen so that just knowing
E
(
x
)
(without knowing the decryption function
D
) doesn’t allow one to discover
anything about the original message
x
.
For centuries cryptography was based on what are now called
privatekey
protocols. In such a scheme,
Alice and Bob meet beforehand and together choose a secret codebook, with which they encrypt all future
correspondence between them. (This codebook plays the role of the functions
E
and
D
above.) Eve’s only
hope then is to collect some encrypted messages and use them to at least partially ﬁgure out the codebook.
Publickey
schemes, such as RSA, are signiﬁcantly more subtle and tricky, but simultaneously also more
useful: they allow Alice to send Bob a message without ever having met him before! This almost sounds
impossible, because in this scenario there is a symmetry between Bob and Eve: why should Bob have any
advantage over Eve in terms of being able to understand Alice’s message? The central idea behind the RSA
cryptosystem is that Bob is able to implement a digital lock, to which only he has the key. Now by making
this digital lock public, he gives Alice (or, indeed, anybody else) a way to send him a secure message which
only he can open. Intuitively, Alice (or anyone) can “apply the lock” to her message, but thereafter only Bob
can “open the lock” and recover the original message.
Here is how the digital lock is implemented in the RSA scheme. Each person has a public key known to the
whole world (the “lock”), and a private key known only to him or herself (the “key to the lock”). When
Alice wants to send a message
x
to Bob, she encodes it using Bob’s public key. Bob then decrypts it using
his private key, thus retrieving
x
. Eve is welcome to see as many encrypted messages for Bob as she likes,
but she will not be able to decode them (under certain simple assumptions explained below).
The RSA scheme is based heavily on modular arithmetic. Let
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 Fall '11
 Rau

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