Unformatted text preview: Ï† (p1). 6) A Carmichael number n is a composite number such that for all values of b with gcd(b,n) = 1, b n1 â‰¡ 1 (mod n). Prove: If n=p 1 p 2 ...p k where each p i for 1 â‰¤ i â‰¤ k is a distinct prime that satisfies (p i 1)  (n 1) for all i, then n is a Carmichael number. 7) We proved that for RSA encryption, where e(x) = x e mod n and d(y) = y d mod n, that d(e(x)) = x if gcd(n,x)=1. Now prove that this statement is true for all x, even if gcd(n,x)>1. 8) Supposed three users in a network, say Angie, Bob and Carmen, all have public encryption exponents e = 3. Let their moduli be denoted by n A , n B , and n C . Now suppose David encrypts the same plaintext x to send to Angie, Bob and Carmen. That is, Alice computes y A = x 3 mod n A , y B = x 3 mod n B , and y C = x 3 mod n C . Describe how Oscar can compute x, given y A , y B and y C . 9) Do 9.10 in the textbook (page 281)....
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 Summer '09

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