The unobfuscatable programs

Barak et al. constructed a family of unobfuscatable programs, for which an efficient attacker can always learn more from any obfuscated code than from black-box access.^[2][3]

Broadly, they start by engineering a special pair of programs that cannot be obfuscated together. For some randomly selected strings ${\displaystyle \alpha ,\beta }$ of a fixed, pre-determined length ${\displaystyle k}$, define one program to be one that computes

${\displaystyle C_{\alpha ,\beta }(x):={\begin{cases}\beta &{\text{if }}x=\alpha \\0&{\text{otherwise}}\end{cases}}}$

and the other program to one that computes

${\displaystyle D_{\alpha ,\beta }(X):={\begin{cases}1&{\text{if }}X(\alpha )=\beta {\text{ and }}X{\text{ runs in time}}\leq {\text{poly}}(k)\\0&{\text{otherwise}}\end{cases}}.}$

(Here, ${\displaystyle D_{\alpha ,\beta }}$ interprets its input as the code for a Turing machine. The second condition in the definition of ${\displaystyle D_{\alpha ,\beta }}$ is to prevent the function from being uncomputable.)

If an efficient attacker only has black-box access, Barak et al. argued, then the attacker only has an exponentially small chance of guessing the password ${\displaystyle \alpha }$, and so cannot distinguish the pair of programs from a pair where ${\displaystyle C_{\alpha ,\beta }}$ is replaced by some program ${\displaystyle Z}$ that always outputs "0". However, if the attacker has access to any obfuscated implementations ${\displaystyle C'_{\alpha ,\beta },D'_{\alpha ,\beta }}$ of ${\displaystyle C_{\alpha ,\beta },D_{\alpha ,\beta }}$, then the attacker will find ${\displaystyle D'_{\alpha ,\beta }(C'_{\alpha ,\beta })=1}$ with probability 1, whereas the attacker will always find ${\displaystyle D'_{\alpha ,\beta }(Z)=0}$ unless ${\displaystyle \beta =0}$ (which should happen with negligible probability). This means that the attacker can always distinguish the pair ${\displaystyle (C'_{\alpha ,\beta },D'_{\alpha ,\beta })}$ from the pair ${\displaystyle (Z,D'_{\alpha ,\beta })}$ with obfuscated code access, but not black-box access. Since no obfuscator can prevent this attack, Barak et al. conclude that no black-box obfuscator for pairs of programs exists.^[2][3]

To conclude the argument, Barak et al. define a third program to implement the functionality of the two previous:

${\displaystyle F_{\alpha ,\beta }(b,x):={\begin{cases}C_{\alpha ,\beta }(x)&{\text{if }}b=0\\D_{\alpha ,\beta }(x)&{\text{if }}b=1\\\end{cases}}.}$

Since equivalently efficient implementations of ${\displaystyle C_{\alpha ,\beta },D_{\alpha ,\beta }}$ can be recovered from one of ${\displaystyle F_{\alpha ,\beta }}$ by hardwiring the value of ${\displaystyle b}$, Barak et al. conclude that ${\displaystyle F_{\alpha ,\beta }}$ cannot be obfuscated either, which concludes their argument.^[2]

Impossible variants of black-box obfuscation and other types of unobfuscable programs

In their paper, Barak et al. also prove the following (conditional to appropriate cryptographic assumptions):^[2]

• There are unobfuscatable circuits.
• There is no black-box approximate obfuscator.
• There are unobfuscatable, secure, probabilistic private-key cryptosystems.
• There are unobfuscatable, secure, deterministic digital signature schemes.
• There are unobfuscatable, secure, deterministic message authentication schemes.
• There are unobfuscatable, secure pseudorandom functions.
• For many protocols that are secure in the random oracle model, the protocol becomes insecure if the random oracle is replaced with an artificial cryptographic hash function; in particular, Fiat-Shamir schemes can be attacked.
• There are unobfuscatable circuits in TC0 (that is, constant-depth threshold circuits).
• There are unobfuscatable sampling algorithms (in fact, these cannot be obfuscated approximately).
• There is no secure software watermarking scheme.