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Unformatted text preview: RSA: Decryption If you get an encrypted message C = M e mod n , how do you decrypt Compute C d M ed (mod n ). Can do this quickly using fast exponentiation again Claim: M ed M (mod n ) Proof: Since ed 1 (mod ( p- 1)( q- 1)) ed 1 (mod p- 1) and ed 1 (mod q- 1) Since ed = k ( p- 1) + 1 for some k , M ed = ( M p- 1 ) k M M (mod p ) (Fermats Little Theorem) True even if p | M Similarly, M ed M (mod q ) Since p , q , relatively prime, M ed M (mod n ) (The- orem 10). Note: Decryption would be easy for someone who can factor n . RSA depends on factoring being hard! 1 Digital Signatures How can I send you a message in such a way that youre convinced it came from me (and can convince others). Want an analogue of a certified signature Cool observation: To send a message M , send M d (mod n ) where ( n,e ) is my public key Recipient (and anyone else) can compute ( M d ) e M (mod n ), since M is public No one else could have sent this message, since no one else knows d . 2 Probabilistic Primality Testing RSA requires really large primes. This requires testing numbers for primality. Although there are now polynomial tests, the stan- dard approach now uses probabilistic primality tests Main idea in probabilistic primality testing algorithm: Choose b between 1 and n at random Apply an easily computable (deterministic) test T ( b,n ) such that T ( b,n ) is true (for all b ) if n is prime. If n is composite, there are lots of b s for which T ( b,n ) is false Example: Compute gcd( b,n ). If n is prime, gcd( b,n ) = 1 If n is composite, gcd( b,n ) negationslash = 1 for some b s Problem: there may not be that many witnesses 3 Example: Compute b n- 1 mod n If n is prime b n- 1 1 (mod n ) (Fermat) Unfortunately, there are some composite numbers n such that b n- 1 1 (mod n ) These are called Carmichael numbers There are tests T ( b,n ) with the property that T ( b,n ) = 1 for all b if n is prime T ( b,n ) = 0 for at least 1 / 3 of the b s if n is composite T ( b,n ) is computable quickly (in polynomial time) Constructing T requires a little more number theory Beyond the scope of this course. Given such a test T , its easy to construct a probabilistic primality test: Choose 100 (or 200) b s at random Test T ( b,n ) for each one If T ( b,n ) = 0 for any b , declare b composite This is definitely correct If T ( b,n ) = 1 for all b s you chose, declare n prime This is highly likely to be correct 4 Prelim Coverage Chapter 0: Sets * Operations: union, intersection, complementa- tion, set difference * Proving equality of sets Relations: * reflexive, symmetric, transitive, equivalence re- lations * transitive closure Functions * Injective, surjective, bijective * Inverse function Important functions and how to manipulate them: * exponent, logarithms, ceiling, floor, mod...
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This note was uploaded on 10/09/2008 for the course COM S 280 taught by Professor Kleinberg during the Spring '05 term at Cornell University (Engineering School).
- Spring '05