When given the above definitions of advantage and $-$indistinguishability and asked to solve Problem $2\mathrm{.}3,$

OpenAI's ChatGPT $($using GPT$-4)$ provided the following response.

To prove that in any encryption scheme that satisfies correctness and $lon-$indistinguishability, we

necessarily have $|K|(1-lon)|M|,$ we'll use a proof by contradiction. We'll assume the opposite,

i$.$e and show that this leads to a violation of $lon-$indistinguishability.

Let's begin with the assumptions:

$|K|$ is the size of the key space.

$|M|$ is the size of the message space.

$lon$ is the parameter representing the level of indistinguishability.

We assume that

Now, we will construct an adversary A that distinguishes between two ciphertexts generated from

distinct plaintexts $m_0$ and $m_1$ with an advantage greater than $lon,$ which contradicts $lon-$indistinguishability.

The adversary A works as follows:

A chooses two distinct plaintexts $m_0$ and $m_1$ from the message space $M.$

A generates a random key $k$ from the key space $K.$

A encrypts both $m_0$ and $m_1$ with the same key $k$ to obtain ciphertexts $c_0=$Enc$(k,m_0)$ and

$c_1=$Enc$(k,m_1).$

A returns $(c_0,c_1).$

Now, let's analyze the behavior of $A$ :

If $m_0$ and $m_1$ are chosen randomly, with probability $lon,c_0=c_1$ because A chose the same key

$k$ for encryption. This is because means there must be some keys $k$ that

encrypt both $m_0$ and $m_1$ with the same ciphertext.

If $m_0$ and $m_1$ are chosen such that they have different ciphertexts under all possible keys

$($which happens with probability $1-lon),$ then $c_0$ and $c_1$ will be different.

Now, let's calculate the advantage of the adversary $A$ :

$Ad{v}_A=|Pr[A$ outputs $1$ when $c_0=c_1]-Pr[A$ outputs $1$ when $c_0{c}_1]|$

$=|lon-0|=lon.$

Since $lon>0,$ this means that A can distinguish between the ciphertexts with an advantage of

$lon,$ contradicting the assumption of $lon-$indistinguishability $($where $lon$ should be an upper bound on the

advantage$).$ Hence, our assumption that must be incorrect.

Therefore, in any encryption scheme that satisfies correctness and $lon-$indistinguishability, we nec$-$

essarily have $|K|(1-lon)|M|.$