Indistinguishability obfuscation

Let ${\displaystyle {\mathcal {iO}}}$ be some uniform probabilistic polynomial-time algorithm. Then ${\displaystyle {\mathcal {iO}}}$ is called an indistinguishability obfuscator if and only if it satisfies both of the following two statements:[5][6][7]

• Completeness or Functionality: For any Boolean circuit C of input length n and input ${\displaystyle x\in \{0,1\}^{n}}$, we have
  $${\displaystyle \Pr[C'(x)=C(x):C'\leftarrow {\mathcal {iO}}(C)]=1.}$$

  
• Indistinguishability: For every pair of circuits ${\displaystyle C_{0},C_{1}}$ of the same size k that implement the same functionality, the distributions ${\displaystyle \{{\mathcal {iO}}(C_{0})\}}$ and ${\displaystyle \{{\mathcal {iO}}(C_{1})\}}$ are computationally indistinguishable. In other words, for any probabilistic polynomial-time adversary A, there is a negligible function ${\displaystyle \varepsilon (k)}$ (i.e., a function that eventually grows slower than ${\displaystyle 1/p(k)}$ for any polynomial p) such that, for every pair of circuits ${\displaystyle C_{0},C_{1}}$ of the same size k that implement the same functionality, we have
  $${\displaystyle |\Pr[A({\mathcal {iO}}(C_{0}))=1]-\Pr[A({\mathcal {iO}}(C_{1}))=1]|\leq \varepsilon (k).}$$

  

In 2001, Barak et al., showing that black-box obfuscation is impossible, also proposed the idea of an indistinguishability obfuscator, and constructed an inefficient one.[8][7][2]Although this notion seemed relatively weak, Goldwasser and Rothblum (2007) showed that an efficient indistinguishability obfuscator would be a best-possible obfuscator, and any best-possible obfuscator would be an indistinguishability obfuscator.[8][9] (However, for inefficient obfuscators, no best-possible obfuscator exists unless the polynomial hierarchy collapses to the second level.[9])

An open-source software implementation of an IO candidate was created in 2015.[10]

It is useful to divide the question of the existence of iO by using Russell Impagliazzo's "five worlds",[13] which are five different hypothetical situations about average-case complexity:[6]

• Algorithmica: In this case P = NP, but iO exists.
• Heuristica: In this case NP problems are easy on average; iO does not exist.
• Pessiland: In this case, ${\displaystyle BPP\neq NP}$, but one-way functions do not exist; as a result, iO does not exist.
• Minicrypt: In this case, one-way functions exist, but secure public-key cryptography does not; iO does not exist (because explicit constructions of public-key cryptography from iO and one-way functions are known).
• Cryptomania: In this case, secure public-key cryptography exists; but iO does not exist.
• Obfustopia:[14][15] In this case, iO is believed to exist.

Indistinguishability obfuscators, if they exist, could be used for an enormous range of cryptographic applications, so much so that it has been referred to as a "central hub" for cryptography,[1][3] the "crown jewel of cryptography",[3] or "crypto-complete".[2] Concretely, an indistinguishability obfuscator (with the additional assumption of the existence of one-way functions[2]) could be used to construct the following kinds of cryptography:

• Indistinguishability obfuscation for programs in the RAM model[5] and for Turing machines[16][17]
• IND-CCA-secure public-key cryptography[18]
• Short digital signatures[18]
• IND-CCA-secure key encapsulation schemes[18]
• Perfectly zero-knowledge non-interactive zero-knowledge proofs and succinct non-interactive arguments[18]
• Constant-round concurrent zero-knowledge protocols[5]
• Multilinear maps with bounded polynomial degrees[5]
• Injective trapdoor functions[18]
• Fully homomorphic encryption[5]
• Witness encryption[5][2]
• Functional encryption[2]
• Secret sharing for any monotone NP language[5]
• Semi-honest oblivious transfer[18]
• Deniable encryption (both sender-deniable and fully-deniable)[18][5][2]
• Multiparty, non-interactive key exchange[5]
• Adaptively secure succinct garbled RAM[5]
• Correlation intractable functions[5]
• Attribute-based encryption[5]
• Oblivious transfer[2]
• Traitor tracing[2]
• Graded encoding schemes[2]

Additionally, if iO and one-way functions exist, then problems in the PPAD complexity class are provably hard.[5][19]

However, indistinguishability obfuscation cannot be used to construct every possible cryptographic protocol: for example, no black-box construction can convert an indistinguishability obfuscator to a collision-resistant hash function family, even with a trapdoor permutation, except with an exponential loss of security.[20]

• Black-box obfuscation, a stronger form of obfuscation proven to be impossible