Question

Suppose that Alice's RSA public key is (N, e) = (33,3) and her private

key is d = 7.
a. If Bob encrypts the message M = 19 using Alice's public key, what
is the ciphertext CI Show that Alice can decrypt C to obtain M.
b. Let S be the result when Alice digitally signs the message M = 25. What is 5? If Bob receives M and S, explain the process Bob will use to verify the signature and show that in this particular case, the signature verification succeeds.

Answer 1

Explanation:

Given Alice's RSA public key (N, e) = (33,3) and Alice's private key is d = 7.

a)

To encrypt: Formula C= M^e mod n = 19^3 = 28 (mod 33) therefore C = 28

To decrypt: Formula M = C^d mod n = 28^7 = 19 (mod 33) therefore M = 19 getting back our original answer proving Alice can decrypt C to obtain M

b) The signed result is Formula S = M^d (mod N) = 25^7 (mod 33) = 31. To verify the signature, Bob computes Formula

S^e (mod N) = 31^3 (mod 33) = 25 therefore the signature is verified as the result matches the received value, M which was 25.