skip to main content

Title: Indistinguishability obfuscation from well-founded assumptions.
Indistinguishability obfuscation, introduced by [Barak et. al. Crypto2001], aims to compile programs into unintelligible ones while preserving functionality. It is a fascinating and powerful object that has been shown to enable a host of new cryptographic goals and beyond. However, constructions of indistinguishability obfuscation have remained elusive, with all other proposals relying on heuristics or newly conjectured hardness assumptions. In this work, we show how to construct indistinguishability obfuscation from subexponential hardness of four well-founded assumptions. We prove: Informal Theorem: Let 𝜏∈ (0,∞), π›Ώβˆˆ (0,1), πœ–βˆˆ (0,1) be arbitrary constants. Assume sub-exponential security of the following assumptions: - the Learning With Errors (LWE) assumption with subexponential modulus-to-noise ratio 2^{π‘˜^πœ–} and noises of magnitude polynomial in π‘˜,where π‘˜ is the dimension of the LWE secret, - the Learning Parity with Noise (LPN) assumption over general prime fields Z𝑝 with polynomially many LPN samples and error rate 1/β„“^𝛿 ,where β„“ is the dimension of the LPN secret, - the existence of a Boolean Pseudo-Random Generator (PRG) in NC0 with stretch 𝑛^{1+𝜏}, where 𝑛 is the length of the PRG seed, - the Decision Linear (DLIN) assumption on symmetric bilinear groups of prime order. Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security more » of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits. « less
; ;
Award ID(s):
Publication Date:
Journal Name:
Sponsoring Org:
National Science Foundation
More Like this
  1. In this work, we study the question of what set of simple-to-state assumptions suffice for constructing functional encryption and indistinguishability obfuscation (IO), supporting all functions describable by polynomial-size circuits. Our work improves over the state-of-the-art work of Jain, Lin, Matt, and Sahai (Eurocrypt 2019) in multiple dimensions. New Assumption: Previous to our work, all constructions of IO from simple assumptions required novel pseudorandomness generators involving LWE samples and constant-degree polynomials over the integers, evaluated on the error of the LWE samples. In contrast, Boolean pseudorandom generators (PRGs) computable by constant-degree polynomials have been extensively studied since the work of Goldreichmore »(2000). (Goldreich and follow-up works study Boolean pseudorandom generators with constant-locality, which can be computed by constant-degree polynomials.) We show how to replace the novel pseudorandom objects over the integers used in previous works, with appropriate Boolean pseudorandom generators with sufficient stretch, when combined with LWE with binary error over suitable parameters. Both binary error LWE and constant degree Goldreich PRGs have been a subject of extensive cryptanalysis since much before our work and thus we back the plausibility of our assumption with security against algorithms studied in context of cryptanalysis of these objects. New Techniques: we introduce a number of new techniques: – We show how to build partially-hiding public-key functional encryption, supporting degree-2 functions in the secret part of the message, and arithmetic NC1 functions over the public part of the message, assuming only standard assumptions over asymmetric pairing groups. – We construct single-ciphertext secret-key functional encryption for all circuits with linear key generation, assuming only the LWE assumption. Simplification: Unlike prior works, our new techniques furthermore let us construct public-key functional encryption for polynomial-sized circuits directly (without invoking any bootstrapping theorem, nor transformation from secret-key to public key FE), and based only on the polynomial hardness of underlying assumptions. The functional encryption scheme satisfies a strong notion of efficiency where the size of the ciphertext grows only sublinearly in the output size of the circuit and not its size. Finally, assuming that the underlying assumptions are subexponentially hard, we can bootstrap this construction to achieve iO.« less
  2. We show the existence of indistinguishability obfuscators (iO) for general circuits assuming subexponential security of: (a) the Learning with Errors (LWE) assumption (with subexponential modulusto- noise ratio); (b) a circular security conjecture regarding the Gentry- Sahai-Waters’ (GSW) encryption scheme and a Packed version of Regev’s encryption scheme. The circular security conjecture states that a notion of leakage-resilient security, that we prove is satisfied by GSW assuming LWE, is retained in the presence of an encrypted key-cycle involving GSW and Packed Regev.
  3. In a traitor tracing (TT) system for n users, every user has his/her own secret key. Content providers can encrypt messages using a public key, and each user can decrypt the ciphertext using his/her secret key. Suppose some of the n users collude to construct a pirate decoding box. Then the tracing scheme has a special algorithm, called 𝖳𝗋𝖺𝖼𝖾 , which can identify at least one of the secret keys used to construct the pirate decoding box. Traditionally, the trace algorithm output only the β€˜index’ associated with the traitors. As a result, to use such systems, either a central mastermore »authority must map the indices to actual identities, or there should be a public mapping of indices to identities. Both these options are problematic, especially if we need public tracing with anonymity of users. Nishimaki, Wichs, and Zhandry (NWZ) [Eurocrypt 2016] addressed this problem by constructing a traitor tracing scheme where the identities of users are embedded in the secret keys, and the trace algorithm, given a decoding box D, can recover the entire identities of the traitors. We call such schemes β€˜Embedded Identity Traitor Tracing’ schemes. NWZ constructed such schemes based on adaptively secure functional encryption (FE). Currently, the only known constructions of FE schemes are based on nonstandard assumptions such as multilinear maps and iO. In this work, we study the problem of embedded identities TT based on standard assumptions. We provide a range of constructions based on different assumptions such as public key encryption (PKE), bilinear maps and the Learning with Errors (LWE) assumption. The different constructions have different efficiency trade offs. In our PKE based construction, the ciphertext size grows linearly with the number of users; the bilinear maps based construction has sub-linear (π‘›βˆš ) sized ciphertexts. Both these schemes have public tracing. The LWE based scheme is a private tracing scheme with optimal ciphertexts (i.e., log(𝑛)). Finally, we also present other notions of traitor tracing, and discuss how they can be build in a generic manner from our base embedded identity TT scheme.« less
  4. We study several strengthening of classical circular security assumptions which were recently introduced in four new lattice-based constructions of indistinguishability obfuscation: Brakerski-Dottling-Garg-Malavolta (Eurocrypt 2020), Gay-Pass (STOC 2021), Brakerski-Dottling-Garg-Malavolta (Eprint 2020) and Wee-Wichs (Eprint 2020). We provide explicit counterexamples to the 2-circular shielded randomness leakage assumption w.r.t. the Gentry-Sahai-Waters fully homomorphic encryption scheme proposed by Gay-Pass, and the homomorphic pseudorandom LWE samples conjecture proposed by Wee-Wichs. Our work suggests a separation between classical circular security of the kind underlying un-levelled fully-homomorphic encryption from the strengthened versions underlying recent iO constructions, showing that they are not (yet) on the same footing. Ourmore »counterexamples exploit the flexibility to choose specific implementations of circuits, which is explicitly allowed in the Gay-Pass assumption and unspecified in the Wee-Wichs assumption. Their indistinguishabilty obfuscation schemes are still unbroken. Our work shows that the assumptions, at least, need refinement. In particular, generic leakage-resilient circular security assumptions are delicate, and their security is sensitive to the specific structure of the leakages involved.« less
  5. In this work, we study the fascinating notion of output-compressing randomized encodings for Turing Machines, in a shared randomness model. In this model, the encoder and decoder have access to a shared random string, and the efficiency requirement is, the size of the encoding must be independent of the running time and output length of the Turing Machine on the given input, while the length of the shared random string is allowed to grow with the length of the output. We show how to construct output-compressing randomized encodings for Turing machines in the shared randomness model, assuming iO for circuitsmore »and any assumption in the set {LWE, DDH, Nπ‘‘β„Ž Residuosity}. We then show interesting implications of the above result to basic feasibility questions in the areas of secure multiparty computation (MPC) and indistinguishability obfuscation (iO): 1.Compact MPC for Turing Machines in the Random Oracle Model. In the context of MPC, we consider the following basic feasibility question: does there exist a malicious-secure MPC protocol for Turing Machines whose communication complexity is independent of the running time and output length of the Turing Machine when executed on the combined inputs of all parties? We call such a protocol as a compact MPC protocol. HubΓ‘cek and Wichs [HW15] showed via an incompressibility argument, that, even for the restricted setting of circuits, it is impossible to construct a malicious secure two party computation protocol in the plain model where the communication complexity is independent of the output length. In this work, we show how to evade this impossibility by compiling any (non-compact) MPC protocol in the plain model to a compact MPC protocol for Turing Machines in the Random Oracle Model, assuming output-compressing randomized encodings in the shared randomness model. 2. Succinct iO for Turing Machines in the Shared Randomness Model. In all existing constructions of iO for Turing Machines, the size of the obfuscated program grows with a bound on the input length. In this work, we show how to construct an iO scheme for Turing Machines in the shared randomness model where the size of the obfuscated program is independent of a bound on the input length, assuming iO for circuits and any assumption in the set {LWE, DDH, Nπ‘‘β„Ž Residuosity}.« less