CSCI6268L13 - 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 #13 Sep 30 2009 CSCI 6268/TLEN 5550, Fall 2009
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Prime Number Theorem Are there enough primes? There are plenty, as exhibited by the PNT: PNT: π (n) is asymptotic to n/ln(n) where π (n) is the number of primes smaller than n In other words, lim n →∞ π (n) ln(n)/n = 1 What does this mean? Primes get sparser as we go to the right on the number line
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π( n) versus n/ln(n)
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Sample Calculation Let’s say we’re generating an RSA modulus and we need two 512-bit primes This will give us a 1024-bit modulus n Let’s generate the first prime, p Question: if I start at some random 512-bit odd candidate c, what is the probability that c is prime? Ans: about 1/ln(c) ≈ 1/350 Question: what is the expected number of candidates I have to test before I find a prime, assuming I try every odd starting from c? Ans: each number has a 1/350 chance, but I’m testing only odd numbers, so my chance is 1/175; I therefore expect to test 175 numbers on average before I find a prime Of course I could do more sieving (eliminate multiples of 3, 5, etc)
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Digital Signatures Digital Signatures are authentication in the asymmetric key model MAC was in the symmetric key model Once again, Alice wants to send an authenticated message to Bob This time they don’t share a key The security definition is the same ACMA model
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We Can Use RSA to Sign RSA gives us a signing primitive as well Alice generates her RSA keys Signing key sk = (d,n) Verification key vk = (e,n) Distributes verification key to the world Keeps signing key private To sign message M Z n * Alice computes sig = M d mod n Alice sends (M, sig) to Bob To verify (M’, sig’) Bob checks to ensure M’ = sig’ e mod n If not, he rejects Once again, don’t do this; use PSS or similar
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Efficiency Why is this inefficient? Signature is same size as message! For MACs, our tag was small… that was good Hash-then-sign We normally use a cryptographic hash function on the message, then sign the hash This produces a much smaller signature 2 nd -preimage resistance is key here Without 2 nd -preimage resistance, forgeries would be possible by attacking the hash function
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Let’s Sum Up Symmetric Key Model Encryption ECB (bad), CBC, CTR All these are modes of operation built on a blockcipher Authentication (MACs) CBC MAC, XCBC, UMAC, HMAC Asymmetric Key Model Encryption RSA-OAEP Assumes factoring product of large primes is hard Authentication RSA signatures Usually hash-then-sign
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Next Up: SSL Next we’ll look at how to put all this together to form a network security protocol We will use SSL/TLS as our model since it’s ubiquitous But first, we’ll digress to talk about OpenSSL, and our first part of the project (a warm-up)
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OpenSSL Was SSLeay Open Source
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