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Annual International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2021: Advances in Cryptology – EUROCRYPT 2021 pp 97–126Cite as

Indistinguishability Obfuscation from Simple-to-State Hard Problems: New Assumptions, New Techniques, and Simplification

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12698))

Abstract

In this work, we study the question of what set of simple-to-state assumptions suffice for constructing functional encryption and indistinguishability obfuscation (\(i\mathcal {O}\)), 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 \(i\mathcal {O}\) 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 Goldreich (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 \(\mathsf {NC^1}\) 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 \(i\mathcal {O}\).

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Notes

  1. 1.

    Throughout this work, unless specified, by degree of boolean PRGs, we mean the degree of the polynomial computing the PRG over the reals.

  2. 2.

    As mentioned in the introduction, partially hiding functional encryption allows to further strengthen the function class supported, by essentially adding computation on a public input, however computation on the private input is still limited to degree 2.

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Acknowledgements

Aayush Jain was partially supported by grants listed under Amit Sahai, a Google PhD fellowship. Huijia Lin was supported by NSF grants CNS-1528178, CNS-1929901, CNS-1936825 (CAREER), the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) under Contract No. W911NF-15-C-0236, and a subcontract No. 2017-002 through Galois.

Amit Sahai was supported in part from DARPA SAFEWARE and SIEVE awards, NTT Research, NSF Frontier Award 1413955, and NSF grant 1619348, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through Award HR00112020024 and the ARL under Contract W911NF-15-C- 0205.

The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, DARPA, ARO, Simons, Intel, Okawa Foundation, ODNI, IARPA, DIMACS, BSF, Xerox, the National Science Foundation, NTT Research, Google, or the U.S. Government.

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  1. IBM, Zurich, Switzerland

    Romain Gay

  2. UCLA, Los Angeles, CA, 90095, USA

    Aayush Jain & Amit Sahai

  3. University of Washington, Seattle, WA, 98195, USA

    Huijia Lin

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  1. Romain Gay
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  1. Inria, Paris, France

    Anne Canteaut

  2. UCLouvain, Louvain-la-Neuve, Belgium

    François-Xavier Standaert

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Gay, R., Jain, A., Lin, H., Sahai, A. (2021). Indistinguishability Obfuscation from Simple-to-State Hard Problems: New Assumptions, New Techniques, and Simplification. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12698. Springer, Cham. https://doi.org/10.1007/978-3-030-77883-5_4

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  • DOI: https://doi.org/10.1007/978-3-030-77883-5_4

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