n6 - CS 70 Discrete Mathematics and Probability Theory Fall...

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CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Note 6 Public-Key Cryptography, RSA, and Modular Arithmetic This lecture, we’ll discuss a beautiful and important application of modular arithmetic: the RSA public-key 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 private-key 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. Public-key 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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n6 - CS 70 Discrete Mathematics and Probability Theory Fall...

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