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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2021: Advances in Cryptology – ASIACRYPT 2021 pp 250–280Cite as

Beyond Software Watermarking: Traitor-Tracing for Pseudorandom Functions

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13092))

Abstract

Software watermarking schemes allow a user to embed an identifier into a piece of code such that the resulting program is nearly functionally-equivalent to the original program, and yet, it is difficult to remove the identifier without destroying the functionality of the program. Such schemes are often considered for proving software ownership or for digital rights management. Existing constructions of watermarking have focused primarily on watermarking pseudorandom functions (PRFs).

   In this work, we revisit the definitional foundations of watermarking, and begin by highlighting a major flaw in existing security notions. Existing security notions for watermarking only require that the identifier be successfully extracted from programs that preserve the exact input/output behavior of the original program. In the context of PRFs, this means that an adversary that constructs a program which computes a quarter of the output bits of the PRF or that is able to distinguish the outputs of the PRF from random are considered to be outside the threat model. However, in any application (e.g., watermarking a decryption device or an authentication token) that relies on PRF security, an adversary that manages to predict a quarter of the bits or distinguishes the PRF outputs from random would be considered to have defeated the scheme. Thus, existing watermarking schemes provide very little security guarantee against realistic adversaries. None of the existing constructions of watermarkable PRFs would be able to extract the identifier from a program that only outputs a quarter of the bits of the PRF or one that perfectly distinguishes.

   To address the shortcomings in existing watermarkable PRF definitions, we introduce a new primitive called a traceable PRF. Our definitions are inspired by similar definitions from public-key traitor tracing, and aim to capture a very robust set of adversaries: namely, any adversary that produces a useful distinguisher (i.e., a program that can break PRF security), can be traced to a specific identifier. We provide a general framework for constructing traceable PRFs via an intermediate primitive called private linear constrained PRFs. Finally, we show how to construct traceable PRFs from a similar set of assumptions previously used to realize software watermarking. Namely, we obtain a single-key traceable PRF from standard lattice assumptions and a fully collusion-resistant traceable PRF from indistinguishability obfuscation (together with injective one-way functions).

R. Goyal—Part of this work was done while at UT Austin and the Simons Institute for the Theory of Computing. Research supported in part by an IBM PhD fellowship and the Simons-Berkeley research fellowship.

S. Kim—Part of this work was done at the Simons Institute for the Theory of Computing. Research supported by NSF, DARPA, a grant from ONR, and the Simons Foundation.

B. Waters—Research supported by NSF CNS-1908611, a Simons Investigator award, and a Packard Foundation Fellowship.

D. J. Wu—Part of this work was done at the University of Virginia and while visiting the Simons Institute for the Theory of Computing. Research supported by NSF CNS-1917414, CNS-2045180, and a Microsoft Research Faculty Fellowship.

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Notes

  1. 1.

    In some cases (e.g., [CHN+16]), the tracing algorithm can still partially recover the identity (e.g., a quarter of the bits) from a circuit that outputs a quarter the bits of each output. But this tracing algorithm can be defeated by an adversary which outputs a circuit that only distinguishes the output of the PRF (i.e., on input (x, y), output 1 if \(\mathsf {Eval}(\mathsf {msk}, x) = y\) and 0 otherwise) or a circuit that computes the parity of the bits of the PRF output.

  2. 2.

    We cannot stipulate that \(T = S\) since the adversary might not use every compromised key when constructing the distinguisher D. The tracing algorithm can only recover the keys the adversary actually uses.

  3. 3.

    A traceable PRF bears many similarities with a constrained PRF [BW13, KPTZ13, BGI14], and all known constructions of collusion-resistant constrained PRFs for sufficiently complex constraints from standard lattice assumptions are secure only in the single-key setting [BV15]. Fully collusion-resistance constrained PRFs for general constraints are only known from indistinguishability obfuscation [BZ14] and one-way functions. Recent work has shown how to construct indistinguishability obfuscation from the combination of multiple standard assumptions [JLS21].

  4. 4.

    As we describe more formally below, the “privacy” requirement refers to a property on the inputs to the PRF, and not the notion of constraint-privacy in the standard definition of a “private constrained PRF” from [BLW17].

  5. 5.

    Throughout the paper, we drop the dependence of spaces \(\mathcal {X}_{\lambda , \kappa }\) and \(\mathcal {Y}_{\lambda , \kappa }\) on security parameter \(\lambda \) and identity length parameter \(\kappa \) whenever clear from context.

  6. 6.

    Note that instead of actually sampling a random function, the challenger simulates it by sampling random input-output pairs on the fly and storing them in a table.

  7. 7.

    As mentioned previously, we drop the dependence on \(\lambda , \kappa \) whenever clear from context.

  8. 8.

    This could also be viewed as a “constrain” algorithm (in the language of constrained PRFs [BW13, KPTZ13, BGI14]), but there are some semantic differences. As such, we refer to this algorithm as a “key-generation” algorithm instead.

  9. 9.

    Here and throughout, the \(\kappa \)-bit identities are interpreted as non-negative integers between 0 and \(2^{\kappa } - 1\) for comparison.

  10. 10.

    Recall that if \(D^{*}\) is a \(\varepsilon \)-good distinguisher, then we have the bound \(\Pr \left[ {D^{*}}^{O_b(\mathsf {msk})}(1^{\lambda }) = b ~ : ~ b \leftarrow \{0,1\} \right] \ge \varepsilon \). This can be rewritten as \(p^{\mathrm {nrml}, D^{*}} - p^{\mathsf {rnd}, D^{*}} \ge 2 \varepsilon \).

  11. 11.

    To implement the punctured programming ideas from [SW14] in the security analysis, we also adjoin a long pseudorandom string to the domain.

  12. 12.

    A puncturable PRF is a constrained PRF (see Sect. 5.1) is a constrained PRF for the family of “puncturing” constraints \(\mathcal {F}= \left\{ f_x :\mathcal {X}\rightarrow \{0, 1\}: x \in \mathcal {X}\right\} \) where \(f_x(y) = 1\) if \(x \ne y\) and 0 if \(x = y\). They can be built directly from one-way functions [GGM84, BW13, KPTZ13, BGI14].

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Authors and Affiliations

  1. MIT, Cambridge, MA, USA

    Rishab Goyal

  2. Stanford University, Stanford, CA, USA

    Sam Kim

  3. University of Texas at Austin, Austin, TX, USA

    Brent Waters & David J. Wu

  4. NTT Research, Sunnyvale, CA, USA

    Brent Waters

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    Mehdi Tibouchi

  2. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Goyal, R., Kim, S., Waters, B., Wu, D.J. (2021). Beyond Software Watermarking: Traitor-Tracing for Pseudorandom Functions. In: Tibouchi, M., Wang, H. (eds) Advances in Cryptology – ASIACRYPT 2021. ASIACRYPT 2021. Lecture Notes in Computer Science(), vol 13092. Springer, Cham. https://doi.org/10.1007/978-3-030-92078-4_9

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