 # 1. Prove the following: If P and q are primes, p t q. p|d, and q|d,...

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1. Prove the following: If P and q are primes, p t q. p|d, and q|d, then pq|d. (I stated this fact without proof in the RSA notes.) 6. Recall our discussion of partial information on RSA. We defined the functions, o parify(2° mod n) = if I mod n is even 1 if I mod n is odd half(2" mod n) o if x mod n < n/2 = 1 if x mod n > n/2 Here e and n are the usual RSA public key parameters, and I is an RSA plaintext message, which means that it's an integer less than n. Prove the following: if half(2° mod n) = 1, then parify((2x) mod n) = 1. 7. Let p be prime. Prove that if I2 III 1 mod P and I # (p - 1) mod p. then 2 III 1 mod p. [Note that (p - 1) III (mod p)] 8. Suppose that p is prime, l' > 0, a III 1 mod p. and god(r.p - 1) = d. Prove that ad III 1 mod p.

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