Codierungstheorie und Kryptographie (SS 2012)
Übung 5
1. Suppose two users Alice and Bob have the same RSA modulus n and
suppose that their encryption exponents eA , eB are relatively prime.
Charles wants to send the message m to Alice and Bob so he encrypts
to get cA ≡ meA mod n and cB ≡ meB mod n. Show how Eve can
find m if she intercepts cA and cB .
2. In the ElGamal cryptosystem, Alice and Bob use p = 31 and g = 3.
Alice chooses her secret key as a = 12 so that A = 8. Alice sends
(c, B) = (5, 2) to Bob. Determine the message m.
3. Let p = 17, q = 19. Suppose you want to encrypt the message x = 27
using RSA algorithm. Choose a suitable encryption exponent e. Then
find the associated decryption exponent d and calculate the encrypted
message y = xe mod n.
4. Use the Baby-Step, Giant-Step algorithm to find the exponent x such
that 3x ≡ 7 mod 31.
5. Use the Miller-Rabin test to conclude that n = 209 is a composite
number. Find one 1 < a < 208 such that the test gives the conclusion
that n = 209 is probably a prime.