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Same Point Composable and Nonmalleable Obfuscated Point Functions

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Applied Cryptography and Network Security (ACNS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12147))

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Abstract

A point obfuscator is an obfuscated program that indicates if a user enters a previously stored password. A digital locker is stronger: outputting a key if a user enters a previously stored password. The real-or-random transform allows one to build a digital locker from a composable point obfuscator (Canetti and Dakdouk, Eurocrypt 2008).

Ideally, both objects would be nonmalleable, detecting adversarial tampering. Appending a non-interactive zero knowledge proof of knowledge adds nonmalleability in the common random string (CRS) model.

Komargodski and Yogev (Eurocrypt, 2018) built a nonmalleable point obfuscator without a CRS. We show a lemma in their proof is false, leaving security of their construction unclear. Bartusek, Ma, and Zhandry (Crypto, 2019) used similar techniques and introduced another nonmalleable point function; their obfuscator is not secure if the same point is obfuscated twice. Thus, there was no composable and nonmalleable point function to instantiate the real-or-random construction.

Our primary contribution is a nonmalleable point obfuscator that can be composed any polynomial number of times with the same point (which must be known ahead of time). Security relies on the assumption used in Bartusek, Ma, and Zhandry. This construction enables a digital locker that is nonmalleable with respect to the input password.

As a secondary contribution, we introduce a key encoding step to detect tampering on the key. This step combines nonmalleable codes and seed-dependent condensers. The seed for the condenser must be public and not tampered, so this can be achieved in the CRS model. The password distribution may depend on the condenser’s seed as long as it is efficiently sampleable. This construction is black box in the underlying point obfuscation.

Nonmalleability for the password is ensured for functions that can be represented as low degree polynomials. Key nonmalleability is inherited from the class of functions prevented by the nonmalleable code.

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Notes

  1. 1.

    We do not consider indistinguishability obfuscation in this work  [GGH+13, GGH+16, SW14, PST14, GLSW15, AJ15].

  2. 2.

    The adversary can always substitute an obfuscation on an unrelated point. Thus, it is possible to create obfuscations for functions f where \(f(\mathsf {val})\) is easy to guess.

  3. 3.

    In the previous sections, we consider X that have worst case min-entropy. However, if \(\tilde{\mathrm {H}}_\infty (X |{\mathsf {seed}}, \mathtt {cond}({\mathsf {seed}}, X ))\ge \beta \) for some \(\beta = \omega (\log \lambda )\) then there exists some \(\beta ' = \omega (\log \lambda )\) such that with \(\Pr _{{\mathsf {seed}}}[\mathrm {H}_\infty (X |{\mathsf {seed}}, \mathtt {cond}({\mathsf {seed}}, X ))\ge \beta ' ]\ge 1- \mathtt {ngl} (\lambda ).\) Thus, this change does not effect the set of distributions assumed to be secure in Assumption 1.

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Acknowledgements

This work was funded in part by a grant from Comcast Inc, by NSF Grant CNS 1849904, and ONR Grant N00014-19-1-2327. This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via Contract No. 2019-19020700008. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.

The authors thank Luke Demarest, Pratyay Mukherjee, Alex Russell, and Mayank Varia for their helpful feedback. Special thanks to James Bartusek, Fermi Ma, and Mark Zhandry for discussing their work and its compositional properties.

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Authors and Affiliations

  1. University of Connecticut, Mansfield, CT, USA

    Peter Fenteany & Benjamin Fuller

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  1. Peter Fenteany
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Correspondence to Benjamin Fuller .

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Editors and Affiliations

  1. Department of Mathematics, University of Padua, Padua, Italy

    Mauro Conti

  2. Singapore University of Technology and Design, Singapore, Singapore

    Jianying Zhou

  3. Dipt di Informatica Sistemi e Produ, Università di Roma “Tor Vergata”, Rome, Roma, Italy

    Emiliano Casalicchio

  4. Sapienza University of Rome, Rome, Italy

    Angelo Spognardi

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Fenteany, P., Fuller, B. (2020). Same Point Composable and Nonmalleable Obfuscated Point Functions. In: Conti, M., Zhou, J., Casalicchio, E., Spognardi, A. (eds) Applied Cryptography and Network Security. ACNS 2020. Lecture Notes in Computer Science(), vol 12147. Springer, Cham. https://doi.org/10.1007/978-3-030-57878-7_7

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  • DOI: https://doi.org/10.1007/978-3-030-57878-7_7

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