There’s lots of similarities in the articles that talk about the impossibility of instantiating the random oracle [CanGolHal98] and obfuscating programs [BGIRSV01]. The biggest takeaway of these stories is that for a long time we (we as in the crypto community) have been thinking about using oracles to abstract away details and make proofs of security easier to deal with. Then, in the real life, one would just have to replace those procedure with an actual implementation of the primitive (like, sha1 or Dual_EC_DRBG). This is totally wrong: there is a substantial difference between giving a program as input and just providing oracle access to it. On the other hand, experience has shown that this idea, although formally wrong, can be a pretty good heuristic.

#### Obfuscation

The topic of obfuscation has been a long-standing quest in crypto, because of its appealing implications, like:

• asymmetric encryption could be reduced to symmetric encryption. (On could public the obfuscation of the symmetric key algorithm with embedded the secret key, and use it as a public key). Interestingly enough, before Diffie-Hellman this was the reason asymmetric encryption was believed to exist.
• any symmetric key cryptosystem can become homomorphic. (Just obfuscate decryption composed with group operation, then make it public)
• software watermarking would become trivial (Just change the program behavior so that it uniquely identifies the user).

We start by formally defining what is an obfuscator, respectively for Turing Machines and circuits.

A probabilistic algorithm $\oracle$ is a TM obfuscator for a family $F$ of Turing Machines if:

• (functionality) $\forall M \in F: [\oracle(M)] = [M]$ (read: they compute the same function);
• (polynomial slowdown) $\forall M \in F: \oracle(M)$ runs in $\mathsf{poly}(\size{M})$
• (virtual black-box property) $\forall$ $\adv$ $\ppt$ distinguisher: $\exists S \in \ppt: \forall M \in F: \abs{\Pr[\adv(\oracle(M)) = 1] - \Pr[S^{\gen{M}}(1^{\size{M}})]} \leq \negl(\size{M}).$

In the above, $\gen{M}$ is the machine that emulates $M$ only for the first $t$ reductions, i.e.:

$\gen{M}: 1^* \to \bin^* \to \bin^*: (1^t, x) \mapsto \begin{cases} y = M(x) &\text{if } M(x) \text{ halts within } t \text{ steps,} \\ \bottom &\text{otherwise}. \end{cases}.$

On the other hand $\negl$ is the family of functions $\NN \to \RR$ that go to zero faster than the inverse of any polynomial, i.e. $\forall c > 0 . \exists k_c: (\negl(k_c) \leq k^{-c} ~~\forall k > k_c)$.

A probabilistic algorithm $O$ is a circuit obfuscator for the following family $F$ of circuits if it satisfies:

• (functionality) as before, $C$ and $\oracle(C)$ compute the same function;
• (polynomial slowdown) as above;
• (virtual black-box property) $\forall \adv ~~\ppt$ distinguisher $\exists S \in \ppt: \forall M \in F \abs{\Pr[\adv(\oracle(C)) = 1] - \Pr[S^{C}(1^{\size{C}}) = 1]} \leq \negl(\size{M}).$

When the family of Turing Machines $F$ is omitted it is assumed to be the collection of all Turing Machines (circuits, respectively).

It is easy to convince ourselves that if a Turing Machine obfuscators exist, then circuit obfuscator exists ([BGIRSV01], Prop 2.3). In some sense, this means that proving impossibility for Turing Machines is easier than for circuits.

One could also wonder why the authors used this virtual black box property instead of something else. In order of aspiration, one could attempt to reach out for:

1. Computational Indistinguishability: The output stream of any $\ppt$ machine given access to $\oracle(P)$ is computationally indistinguishable from the one when given just oracle access.
2. Satisfying a Relation: the adversary produces an output that satisfies some relation $R$ together with the input program (possibly $\poly$-computable relations). The same stream can be produced (with roughly the same probability) when given $\oracle(P)$ as input as when given only oracle access to $P$.
3. Computing a Function: as above, but the relation is a function
4. Satisfying a Predicate: as above, but the function has image in $\bin$.

Turns out this is the smallest possible requirements one could give to obfuscation, and the instatisfiability for all the above definitions is pretty straightfoward. For instance:

Item 2 cannot be satisfied.

Consider the relation $\set{(P, \oracle(P))}_{P \in \ppt}$. It is always satisfied when given as input an obfuscated program, but difficult when given only oracle access.

Everything is still the same as in the above except:

• virtual black-box property: $\forall \adv ~~\ppt$ distinguisher $\exists S \in \ppt: \forall C, D \in F \abs{\Pr[A(\oracle(C), \oracle(D)) = 1] - \Pr[S^{\gen{C, D}}(1^{\size{C} + \size{D}})]} \leq \negl(\min(\size{C}, \size{D}))$

Neither 2-TM obfuscator nor 2-circuit obfuscator do exist.

We start defining some functions that are difficult to learn with oracle queries: given $\alpha, \beta \in \bin^\secpar$ define:

\begin{aligned} C_{\alpha, \beta}(x) &\defeq \begin{cases} \beta & \text{if } x = \alpha \\ 0^\secpar & \text{otherwise}. \end{cases} \\ D_{\alpha, \beta}(C) &\defeq \begin{cases} 1 & \text{if } C(x) = \beta \\ 0 & \text{otherwise.} \end{cases} \\ Z_\secpar(x) &\defeq 0^\secpar \end{aligned}

Note: $D_{\alpha, \beta}$ can be uncomputable, but we can execute $C$ for $\poly$ steps and output $0$ is none was given from the execution of $C$. Now consider the adversary $\adv \in \ppt : \adv(C, D) = D(C)$. Note: also in this case we should let $\adv$ run only for $\poly(\min(\size{C}, \size{D})) \subset \poly(\secpar)$. Then, the following holds:

\begin{aligned} \Pr[\adv(\oracle(C_{\alpha, \beta}), \oracle(D_{\alpha, \beta}))] &\defeq \Pr[\adv_C] = 1 \tag{i} \\ \abs{ \Pr[S^{C_{\alpha, \beta}~~ D_{\alpha, \beta}}(\secparam) =1] - \Pr[S^{Z_\secpar ~~ D_{\alpha, \beta}}(\secparam) = 1]} &\defeq \abs{ \Pr[S_C] - \Pr[S_Z]} \leq 2^{-\Omega(\secpar)} \tag{ii} \\ \Pr[\adv(\oracle(Z_\secpar), \oracle(D_{\alpha, \beta}) = 1] &\defeq \Pr[\adv_Z] = 2^{-\secpar}. \tag{iii} \end{aligned}

By contradiction, if a 2-TM obfuscator existed, we would have that i. $\abs{\Pr[\adv_C] - \Pr[S_C]} \leq \negl(\secpar)$ (for any adversary $\adv$ there exists $S$); ii. $\abs{\Pr[\adv_Z] - \Pr[S_Z]} \leq \negl(\secpar)$ (for any adversary $\adv$ there exists $S$). Therefore:

\begin{aligned} \negl(\secpar) &\geq \abs{\Pr[\adv_C] - \Pr[S_C]} + \abs{\Pr[\adv_Z] - \Pr[S_Z]} \\ &= \abs{\Pr[\adv_C] - \Pr[S_C]} + \abs{\Pr[S_Z] - \Pr[\adv_Z]} \\ &\geq \abs{\Pr[\adv_C] - \Pr[S_C] + \Pr[S_Z] - \Pr[\adv_Z]} \\ &= \abs{1 - 2^\secpar + (\Pr[S_Z] - \Pr[\adv_Z])} \\ &\geq \abs{1 - \negl(\secpar)}, \end{aligned}

which is a contradiction. In all this, we did not prove that $\adv(\oracle(C_\alpha, \beta), \oracle(D_{\alpha, \beta})) \in \poly(\secpar)$, however this follows immediately from the polynomial slowdown property of obfuscators.

The proof for the circuit case is very similar, we just need to make $D_{\alpha, \beta}$ big enough to contain a description of $C$ as input. This This can be solved letting $D_{\alpha, \beta}$ have input length $\poly$ and the proof would work as before.

TM obfuscators do not exist.

Given $f_0, f_1: X \to Y$ define $f_0 \concat f_1: \bin \times X \to Y: (b, x) \mapsto f_b(x)$ the concatenation of two functions (or Turing Machines, or circuits).

Consider $\adv$ to be the adversary that behaves like in the previous proposition but after splitting hte concatenated programs:

$$\adv(C) = \adv(C_0 \concat C_1) = C_1(C_0).$$ Then define $F_{\alpha, \beta} \defeq C_{\alpha, \beta} \concat D_{\alpha, \beta}$ and $G_{\alpha, \beta} \defeq Z_\secpar \concat D_{\alpha, \beta}$. We have that:

\begin{aligned} \Pr[\adv(\oracle(F_{\alpha, \beta}) = 1] &= 1 \tag{i} \\ \abs{ \Pr[S^{F_{\alpha, \beta}}(\secparam)=1] - \Pr[S^{G_{\alpha,\beta}}(\secparam) = 1]} &\leq 2^{-\Omega(\secpar) } \tag{ii} \\ \Pr[\adv(\oracle(G_{\alpha, \beta}) = 1] &= 2^{-k}. \tag{iii} \end{aligned}

In the circuit case, we can’t bring the same argument; $\adv$ on input $\oracle(F_{\alpha, \beta}) = \oracle(C_{\alpha, \beta} \concat D_{\alpha, \beta}) = F_0 \concat F_1$:

• requires $\size{F_1} > \size{F_0}$ as one takes as input the other, but we have no guarantee for this after polynomial slowdown by the obfuscator.
• requires $\size{F_0}= \size{F_1}$ since $F_0 = \oracle(F_{\alpha, \beta})(0, x)$ and $F_1 = \oracle(F_{\alpha, \beta})(1, x)$.

So proving it is a bit of a mess. I have a good summary of the steps that need to be taken, but I still need to put them down here.

Before closing up, the authors attempt to patch up the defition of obfuscator. They define something called indistinguishability-obfuscation (iO for friends), which is one of the nicest unicorns in crypto these days.

An indistinguishability obfuscator is defined as a circuit obfuscator except that the virtual black-box property is reduced to:

• (indistinguishability) $\forall \adv \in \ppt \quad \exists \alpha \in \negl(\secpar)$ such that $\forall C_0, C_1$ circuits computing the same function and having the same size $\size{C_0} = \size{C_1} = \secpar$:
$\abs{\Pr[\adv(\oracle(C_0)) = 1] - \Pr[\adv(\oracle(C_1)) = 1]} \leq \alpha.$

Indistinguishability Obfuscation exists.

Let $\oracle(C)$ be the lexicographically first circuit of size $\size{C}$ that computes the same function as $C$.

#### The Random Oracle Methodology

The random oracle rabbit hole is pretty much of the same type. In here, however, we show that there are relations that one can trivially satisfy when playing with actual implementations and never with oracle access.

In addition to that, the coup de grace of [CanGolHal98] consists in showing the existence of (contrived) protocols that are secure in the Random Oracle model but for which there exist no secure implementation.

Informally, a random oracle $\oracle$ is a machine that upon receiving $x \in \bin^*$:

• if it wasn’t, select a uniformly random element from $\bin^{\ell(\secpar)}$ and output it as $\oracle(x)$.

A length function is a map $\ell: \NN \to \NN$ super-logarithmic and polynomially bounded.

Given $\ell: \NN \to \NN$ a length function, an $\ell$-ensemble is a sequence $F = \set{F_\secpar}_{\secpar \in \NN}$ of families of maps $F_\secpar = \set{f_s: \bin^* \to \bin^{\ell(\secpar)}}_{s \in \bin^\secpar}$ so that there exists $\eval \in \poly(\secpar)$ such that $\forall s, x \in \bin^* . [\eval(s, x)] = f_s(x).$ The string $s$ is called description or seed.

We attempt to define relations that are hard to satisfy in the Random Oracle model:

A relation $R$ is said to be evasive w.r.t. some length function $\ell: \NN \to \NN$ if $\forall M ~~\ppt$ machine with oracle access: $$\Pr_\oracle[x \gets M^\oracle(\secparam) ~\land~ (x, \oracle(x)) \in R] \leq \negl(\secpar)$$

So for instance the relation $(x, 0 \concat y^{\secpar - 1}) \quad\forall x, y$ is not evasive, while $(x, 0^\secpar)$ is so.

If $R$ is a binary relation satisfying: $$\sup_{x \in \bin^*} \left( \Pr_{y \in \bin^{\ell(\secpar)}}[(x, y) \in R] \right) \leq \negl(\secpar)$$ then $R$ is evasive.

\begin{aligned} \Pr_O[x \gets M^\oracle(\secparam), (x, \oracle(x)) \in R] &\leq \poly(\secpar) \sup \left( \Pr_{x \in \bin^*}[(x, \oracle(x)) \in R] \right) \\ &\leq \poly(\secpar) \negl(\secpar) = \negl(\secpar). \end{aligned}

Let $F$ be a $\ell$-ensemble. We say that $F$ is correlation intractable w.r.t. $R$ if $R$ is fixed.

Note: we are considering all binary relation between bitstrings here; a weaker notion could require $R$ to be decided in $\poly(\secpar)$. However, this will not change the impossibility results.

Correlation intractable function ensemble do not exist.

Let $F$ be an $\ell$-ensemble. Consider the following binary relation: $$R^F \defeq \bigcup_{\secpar \in \NN} \set{(s, f_s(s)) | s \in \bin^\secpar}.$$ Clearly, this relation is poly-time decidable since $f_s \in \poly(\secpar)$ and evasive, since:

\begin{aligned} & \forall x \in \bin^\secpar \quad \exists!y \in \bin^{\ell(\secpar)} . (x, y) \in R^F \tag{by definition of function}\\ \implies & \Pr_y[(x, y) \in R^F] \leq 2^{-\ell(\secpar)} \leq \negl(\secpar). \end{aligned}

However, if we consider the machine $\1(x) = x \quad \forall x \in \bin^*$:

\begin{aligned} & \Pr_{s \in \bin^{\secpar}}[x \gets \1(s), (x, f_s(x)) \in R^F] \\ =& \Pr_{s \in \bin^{\secpar}}[(s, f_s(s)) \in R^F] = 1, \end{aligned}

which means that $F$ is not correlation intractable.

The above theorem should be already enough to be convinced of the uninstantiability of the random oracle. In the origial paper [BGIRSV01] the authors prove the existence of a signature scheme that is existentially unforgeable under chosen-message attack in the random oracle model but for any possible choice of $F$ the scheme is insecure.