Problem 1: ElGamal Authentication

Once Happy understood ElGamal signatures, he was excited to use them for authentication. He wants to send an authenticated message m to Bob so that Bob can verify that m came from him.

Happy has an ElGamal signing key (g,p,x) and Bob has the corresponding verification key (g,p,a). We denote the signing algorithm using that key pair by S and the verification algorithm by V . Happy and Bob also have a cryptographic hash function h whose output is the same length as the signatures produced by S.

Here’s Happy’s idea. Bob sends him a fresh tag r. Happy signs r and attaches it to a hash of his message. Bob checks the tag’s signature and accepts the message.

	Happy		Bob
1.			Choose random string r.
2.	Compute s = S(r) ⊕ h(m ⊕ r)		Check V (r,s ⊕ h(m ⊕ r)).
			Accept m as coming from Happy if check succeeds.

Questions

1. Describe why Bob accepts every message that Happy sends in this way (assuming no errors in transmission).
2. Mallory wants to replace m with a message m′ of his choosing and get Bob to accept it as valid. Describe in detail how he can do this. Assume that Mallory is carrying out a man-in-the-middle attack, but she does not know Happy’s signing key and cannot forge signatures S(x) for messages x of Mallory’s choosing.
3. Suggest a way to fix this protocol to thwart Mallory’s attack. Your suggestion should not use any more rounds of communication nor assume any other encryption system or secret keys. Explain.
   [Hint: Think about a better way to use h to “bind” m to the signature.]

Problem 2: Hash from Cryptosystem

Happy decided to build a hash function H(M) out of the AES-128 encryption function E_k.

Define the function f(s,m) = E_m(s) ⊕ m, where s and m have length 128. Let M be a message of arbitrary length. Here’s how to compute H(M).

• Pad M appropriately and divide it into 128-bit blocks m₁m₂…m_t.
• Compute the sequence:

• Define H(M) = s_t.

Questions

1. Given any k ≥ 1 and 128-bit string s_k, show how to find a message M = m₁m₂…m_k such that H(M) = s_k.
   [Hint: Use the fact that the decryption function D_k() is the inverse of E_k(). This allows you to “work backwards” from s_k to s₁.
2. Show how to find a colliding pair (M,M′) for H().