# Does there exist an indistinguishability formal proof obfuscator?

A formal proof obfuscator is a mapping $\mathcal{O}$ such that whenever $P$ is a formal proof of a theorem $T$, then $\mathcal{O}(P)$ is a distribution of formal proofs of the same theorem $T$. An indistinguishability formal proof obfuscator is a formal proof obfuscator $\mathcal{O}$ such that whenever $P_{1},P_{2}$ are proofs of of a theorem $T$ the distributions $\mathcal{O}(P_{1}),\mathcal{O}(P_{2})$ are indistinguishable. Does there exist an indistinguishability formal proof obfuscator? Are there any references for an indistinguishability formal proof obfuscator or any formal proof obfuscator in general?

Since you do not give $T$ as an input, the existence of such an $\mathcal{O}$ seems
like it should significant constraints on the is_a_proof_of relation:

For example, if there are proofs P0,P2,P1 and Theorems T0,T1,T2 such that
[for all elements i and j of {0,1,2}, ​ Pi is_a_proof_of Tj ​ if and only if ​ i ≠ j ]
then there would have to be a P such that P is a proof of all three of those theorems.

For the rest of this answer, assume there is a function theor_of_ such that
for all proofs P and theorems T, if P is a proof of T then ​ theor_of_(P) = T .

Under that assumption, sure. ​ ​ ​ For such a function theor_of_, one can
have $\mathcal{O}$ be given by ​ ​ ​ $\mathcal{O}(P)$ ​ = ​ lexicographically least proof of theor_of_(P) ​ .
That $\mathcal{O}$ is in Ptheor_of_[1],NP.

You probably want $\mathcal{O}$ to be efficient, or at least significantly faster than brute force for suitable input lengths, even if NP is hard. ​ For that requirement, it becomes very significant that

you do not give the security parameter as input
and
you're not relating the lengths of the proofs

.

The distributions thus do not depend on the security parameter, so if they're indistinguishable then they're identical. ​ Efficiency in particular forces output-length to be not so much longer than input length, so for the resulting distributions to be identical, $\mathcal{O}$ must find a proof which is not so much longer than the shortest proof, regardless of how long its input proof is.

The latter is the much stricter requirement - For example, consider statements of the form "There exists an x such that $\big[\big[$x satisfies $\phi \big]$ or [length(x) = L]$\big]$", and proof systems for which the proofs of such statements are exactly ordered pairs of the statement and such an $x$.
By setting L so that it will be slightly greater than
$\big[\mathcal{O}$'s runtime on proofs of the resulting formula for which $x$ is a satisfying assignment$\big]$ (remember that L is only log(L) bits long) and giving $\mathcal{O}$ a proof with an L-bit $x$, $\mathcal{O}$ must solve $\phi$.
Even for proof systems that are not so insistent on constructiveness, one could try
replacing ​ length(x) = L ​ with something else such that $\big[$one also gets a proof but it's
probably hard to find a proof that's not sufficiently longer than assignments to $\phi \big]$.

Even if you [give the security parameter as an input] and [only require indistinguishability against uniform adversaries that must choose equal-length proofs], such an $\mathcal{O}$ for a
similarly-constructive proof system would suffice for inverting candidate one-way functions with probability greater than 1/3 ​ - ​ Just run it on proofs that give r for statements
"There exists an x such that f(x) is in {$\hspace{.02 in}$y,z}.", where a random one of
of y,z is the target image, the other of them is f(r), and r is chosen
uniformly from strings with length equal to the security parameter.

Thus, if one-way functions exist then for a feasible such $\mathcal{O}$ to exist,
the proof system must not be feasibly constructive.

Now, I'll finally give the positive results:

As warmup, if there is an efficient non-interactive witness-indistinguishable proof system
NIWIP for SAT with deterministic verifier, then for axiomatic systems S such that

a theor_of_ function, as described near the top of this answer, is efficiently computable
and
the proof system includes arithmetic and
[can prove enough about its own provability predicate]
and
there is a poly(k,M)-time algorithm for finding a proof in ​ S + Con(S) ​ that
NIWIP with security parameter k is sound for statements of length at most M

then there is an efficient $\mathcal{O}$ that takes as input

the security parameter, in unary
and
a length bound L, in unary
and
a proof in S of a $\Pi_1$ formula

and outputs a proof in ​ S + Con(S) ​ of the same $\Pi_1$ formula,
such that the desired indistinguishability will hold when

the security parameters are equal
and
the length bounds are equal
and
the proofs both satisfy the length bound

.

Furthermore, if NIWIP's indistinguishability is perfect,
then $\mathcal{O}$ does not need the security parameter.

(In either case, use NIWIP on the existence of a proof of length at most L, prove the relevant
soundness property of NIWIP to deduce the existence of a proof in S of the $\Pi_1$ formula, and
then use Con(S) and [provable-in-S facts about being provable-in-S] to deduce the $\Pi_1$ formula.)

It's finally time for the full result. ​ For axiomatic systems S0 such that

a theor_of_ function, as described near the top of this answer, is efficiently computable
and
the proof system includes arithmetic and
[can prove enough about its own provability predicate]

, ​ ​ ​ with the axiomatic systems ​ S1,S2,S3,S4,...,S$_{\infty}$ ​ given by

Sn+1 is Sn plus [the schema with one axiom per wff p, where those
axioms are padded (for example, by ANDing with 0=0) to each be longer
than n bits, and otherwise just assert "if p is provable in Sn then p"]
and
S$_{\infty}$ ​ = ​ S0 + S1 + S2 + S3 + S4 + ...

, if there is $\big[$an efficient non-interactive witness-indistinguishable proof system NIWIP for SAT with deterministic verifier$\big]$ and $\big[$a poly(k,M)-time algorithm for finding a proof in S$_{\infty}$
that NIWIP with security parameter k is sound for statements of length at most M$\big]$,
then then there is an efficient $\mathcal{O}$ that takes as input

the security parameter, in unary
and
a length bound L, in unary
and
a proof in S$_{\infty}$

and outputs a proof, also in S$_{\infty}$, of the same formula,
such that the desired indistinguishability will hold when

the security parameters are equal
and
the length bounds are equal
and
the proofs both satisfy the length bounds

.

As before, if NIWIP's indistinguishability is perfect,
then $\mathcal{O}$ does not need the security parameter.

(In either case, work as follows: ​ ​ ​ Proofs in S$_{\infty}$ of length at most L cannot use axioms that aren't in SL, so use NIWIP on the existence of a proof in S$_{\infty}$. ​ From there, prove the relevant soundness property of NIWIP to deduce the existence of a proof in SL of the relevant formula, and then use the relevant axiom from SL+1 to deduce the relevant formula itself.)

I'm not aware of any candidates for that which Shor's algorithm won't break.