What is the appropriate public key encryption for secure coin flipping?

I quote Bobby's question here since I encountered the same one...

Random Coin Flip using ElGamal and a Trusted Party

Consider the following protocol for two parties to flip a fair coin.

1. Trusted party $T$ publishes her public key $K_{pub}$

2. $A$ chooses a random bit $b_A$, encrypts it under $K_{pub}$, and announces the ciphertext $c_A$ to $B$ and $T$

3. $B$ does the same and announces a ciphertext $c_B$ (where $c_B \neq c_A$)

4. $T$ decrypts $c_A$ and $c_B$, and announces the results. Both parties $XOR$ the results to obtain the random value $b_A\oplus b_B$.

It was shown that ElGamal was not appropriate for the encryption scheme. RSA seems to fail to work also. Since $c_A=m^e \mod N$. Then if $m=1$, $c=1$. If $B$ receives $A$'s cipher he will know the exact value of $m$.

So what should the appropriate encryption scheme be?

• This topic appears to have already been covered in How to fairly select a random number for a game without trusting a third party? and Verify product without revealing multipliers. – D.W. Sep 7 '12 at 8:29
• With the protocol as described, active attacks, and if $T$ accepts to play the protocol several times with the same private key, all kinds of horrors can happen, regardless of the qualities of the encryption scheme. E.g. $B$ plays the protocol normally to learn $b_A$, except $B$ captures the communication from $A$ to $T$ at step 2, and blocks communication from $T$ to $A$ at step 4; $B$ then replay the former step 2 to $T$ pretending to be $A$, makes another step 3 where $B$ choses $b_B=b_A\oplus1$, and leaves step 4 proceed normally. The outcome is always $1$, and $B$ knows that. – fgrieu Dec 14 '15 at 18:20

With the added restriction that $c_a \neq c_b$, any CCA-secure cryptosystem will work. For example, Cramer-Shoup instead of Elgamal or RSA+OAEP instead of textbook RSA.

There is no asymmetric encryption scheme which would work here. Whatever algorithm is used, $B$ can always send the exact same message than $A$ (i.e. set $c_B = c_A$); when $T$ decrypts both messages, he gets the same value both times, and $b_A \oplus b_B$ ends up being zero.

ElGamal encryption just makes it a bit easier for $B$ to game the protocol inconspicuously: $B$ can send the same encrypted bit than $A$ (without knowing it) under a random guise, making it indistinguishable from an honestly encrypted bit. But the core issue is still there: this coin flipping protocol is inherently broken.

• I forget to add that $c_A \neq c_B$. Any improvement if we add the above constraint? Thanks. – Jake Dec 1 '11 at 0:13

Note: I originally posted this as an answer to another question before realizing that it was a duplicate, so I'm reposting it here.

As already noted, any scheme that lets $B$ choose $c_B = c_A$ will allow $B$ to cheat by doing so. More generally, so will any scheme that lets $B$, given $c_A$, choose a valid ciphertext $c_B = f(c_A)$ such that $c_B$ still encrypts the same (or the opposite) bit as $c_A$.

Even more generally, $B$ will have an unfair advantage if (and only if!) there exists any function $f$ such that, given $c_A$ and $c_B = f(c_A)$, $B$ can predict $b_A \oplus b_B$ with better than 50% chance. In fact, this is effectively just a restatement of the problem: $B$ can cheat if and only if they can, given $c_A$, choose $c_B$ so as to have an advantage in predicting $b_A \oplus b_B$.

So, what does it take to prevent this? I will argue below that any IND-CCA2 secure encryption scheme will, by definition, suffice to prevent $B$ from cheating (by showing that, if $B$ could cheat, they could also break the IND-CCA2 security), provided that $B$ is prevented from choosing $c_B = c_A$. (Since the ciphertexts are public, this additional restriction can be easily enforced; without it, the problem has no solution, as $B$ can always cheat.)

Proof: Assume that, given a ciphertext $c_A$ encoding a random bit $b_A \in \{0,1\}$, $B$ can choose a valid ciphertext $c_B \ne c_A$, also encoding a single bit, such that they have a non-negligible advantage in guessing whether $c_B$ encodes the same bit as $c_A$. Then $B$ can also break the IND-CCA2 resistance of the encryption scheme as follows:

1. Submit $M_0 = 0$ and $M_1 = 1$ to the challenger as the two challenge plaintexts.
2. Treat the received challenge ciphertext as $c_A$, and compute the corresponding $c_B$.
3. Submit $c_B$ to the decryption oracle to learn which bit it encodes; this is allowed under the rules of the CCA2 game, since $c_B \ne c_A$.
4. Since $B$ now knows which bit $c_B$ encodes, and (by assumption) has a non-negligible advantage in guessing the XOR of this bit and the challenge bit, they have the same advantage in guessing the challenge bit. $\square$