Game with symmetric key

Question

Alice and Bob are playing Rock-paper-scissors.

Alice chooses $a \leftarrow\{stone, paper, scissors\}$ and a nonce $R_A$ used as symmetric key for encryption

$$A → B : A, R_A(a)$$

Bob chooses $b \leftarrow\{stone,paper, scissors\}$ and nonce $R_B$

$$B → A : B, R_B(b)$$

$$A → B : A, R_A$$

Now Bob decodes $R_A(a)$ and knows a and so he knows who is the winner

$$B → A : B, R_B$$

Now Alice decodes $R_B(b)$ and knows $b$ and so he knows who is the winner

What are the possible fraudulent behaviours the players could engage in?

Answer 1

If Alice and Bob use XOR, then Bob can win in every game. For example, Bob select "paper", Alice select "sciss" (let length will be same). After decryption, Bob realize he lost. Then he send to Alice as a key "paper"^Rb^"stone". After XORing Alice finds "stone", so she will think, she lost.

Answer 2

You don't specify your encryption method. To show you why that is important, I'm going to pick a really bad one. Let's say we use a stream cipher to encrypt. "stone" and "paper" have the same length, so we are good there, but "scissors" is longer. So, if Alice chooses "scissors", Bob can see the ciphertext length and know what she picked and choose "stone". Therefore, if Alice chooses "scissors", Bob always wins.

If you pick your encryption method so that the ciphertext lengths are always the same, this attack is mitigated.

Bob, once he knows whether or not he wins, could refuse to send $R_B$.

The major flaw, however, is, even if you pick a really good encryption method (say AES-CBC with a fixed length size), you are doing nothing for integrity checking when integrity is really the primary concern here.

The proper way to do this would be to have Alice and Bob each publish a commitment to their choice. Then they open the commitment publicly. This could be done, for example, with HMAC. Choose a random key $k$ and publish $HMAC(k,m)$ where $m$ is the choice. Once both have done this, they can each publish $m,k$. That way they can verify that the choice hasn't changed.