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Math135Lecture16StudentNotes-1

Math135Lecture16StudentNotes-1 - Math 135 Lecture 16 RSA...

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Math 135: Lecture 16: RSA Public Key Systems In a each pair of users must possess the same key. In a keys are divided into two parts. A public encryption key is shared in a repository. A private decryption key is held secretly by each participant. For user A to send a private message to user B , A would look up B ’s public key, encrypt the message and send it to B . Since B is the only person who possesses the secret key required for decryption, only B can read the message. Key Distribution How do you manage keys among 200 embassies? In a private key system, users must exchange In a public key system, users must The key distribution problem is solved. RSA The possibility of public key cryptography was first published in 1976 in a paper by Di e, Hellman and Merkle. The RSA scheme, named after its discoverers Rivest, Shamir and Adleman is an example of a commercially implemented public key scheme. Messages Are Integers In RSA, How does one get an integer from plaintext? One possibility: Do what we did with a Vigen` ere cipher, assign a number to each letter of the alphabet and then concatenate the digits together. Example 16.1. With A 00 , B 01 , . . . MATH 1
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Square and Multiply Algorithm 1. To compute M e (mod n ) for large e , write e in binary as e = ( r t ...r 2 r 1 r 0 ) 2 where each r i = 0 OR 1 . 2. Compute M, M 2 , M 4 , M 8 , ..., M 2 t 1 , M 2 t ( mod n ) by squaring the previous term in the sequence. 3. Multiply the appropriate terms together, modulo n, to obtain M e Π M 2 i ( mod n ) . Eg R1 Use the square and multiply algorithm to compute 2 29 (mod 187) . 1. 29 = 16 + 4 + 2 + 1 = ( 11101 ) 2 . 2. Compute 2 , 2 2 , 2 4 , 2 8 , 2 16 (mod 187) .
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