CSCI6268L12 - Foundations of Network and Computer Security...

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Foundations of Network and Foundations of Network and Computer Security Computer Security J J ohn Black Lecture #12 Sep 23 rd 2009 CSCI 6268/TLEN 5550, Fall 2009
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RSA Example Let p = 19, q = 23 These aren’t large primes, but they’re primes! n = 437 φ (n) = 396 – Clearly 5 Z * 396 , so set e=5 Then d=317 ed = 5 x 317 = 1585 = 1 + 4 x 396 pk = (5, 437) sk = (396, 437)
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RSA Example (cont) Suppose M = 100 is Alice’s message Ensure (100,437) = 1 Compute C = 100 5 mod 437 = 85 Send 85 to Bob Bob receives C = 85 Computes 85 317 mod 437 = 100 We’ll discuss implementation issues later
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RSA Proof • Need to show that for any M Z n * , M ed = M mod n ed = 1 mod φ (n) [by def of d] So ed = k φ (n) + 1 [by def of modulus] – So working in Z n * , M ed = M k φ (n) + 1 = M k φ (n) M 1 = (M φ (n) ) k M = 1 k M = M Do you see LaGrange’s Theorem there? This doesn’t say anything about the security of RSA, just that we can decrypt
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Security of RSA Clearly if we can factor efficiently, RSA breaks It’s unknown if breaking RSA implies we can factor Basic RSA is not good encryption There are problems with using RSA as I’ve just described; don’t do it Use a method like OAEP We won’t go into this
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Factoring Technology Factoring Algorithms Try everything up to sqrt(n) Good if n is small Sieving Ditto Quadratic Sieve, Elliptic Curves, Pollard’s Rho Algorithm Good up to about 40 bits Number Field Sieve State of the Art for large composites
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This note was uploaded on 03/11/2010 for the course CS 6268 taught by Professor Black during the Spring '09 term at University of Colombo.

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CSCI6268L12 - Foundations of Network and Computer Security...

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