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Watermarkable Public Key Encryption with Efficient Extraction Under Standard Assumptions

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Security and Cryptography for Networks (SCN 2022)

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Abstract

The current state of the art in watermarked public-key encryption schemes under standard cryptographic assumptions suggests that extracting the embedded message requires either linear time in the number of marked keys or the a-priori knowledge of the marked key employed in the decoder. We present the first scheme that obviates these restrictions in the secret-key marking model, i.e., the setting where extraction is performed using a private extraction key. Our construction offers constant time extraction complexity with constant size keys and ciphertexts and is secure under standard assumptions, namely the Decisional Composite Residuosity Assumption [Eurocrypt’99] and the Decisional Diffie Hellman in prime order subgroups of square higher order residues.

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Notes

  1. 1.

    To see this, consider A DDH challenge for the two underlying groups \(\langle g_1^{q'}, y_\mathsf {l}, G_\mathsf {l}, Y_{\mathsf {l}}\rangle \) \(\langle g_1^{p'}, y_\mathsf {r}, G_\mathsf {r}, Y_{\mathsf {r}}\rangle \), we can combine them to \(\langle g_1, y_\mathsf {l}\cdot y_\mathsf {r}, G_\mathsf {l}\cdot G_\mathsf {r}, Y_{\mathsf {l}}^{q'} \cdot Y_{\mathsf {r}}^{p'} \rangle \). Observe that if the challenge pair is DDH distributed then \(G_\mathsf {l}\cdot G_\mathsf {r} = g_1^{q' r_\mathsf {l}+ p' r_\mathsf {r}} \) and \(Y_\mathsf {l}^{q'}\cdot Y_\mathsf {r}^{p'} = y_\mathsf {l}^{q' r_\mathsf {l}} y_\mathsf {r}^{ p' r_\mathsf {r}} = g_1^{ (q')^2 t_\mathsf {l} r_\mathsf {l} + (p')^2 t_\mathsf {r} r_\mathsf {r} } \). Now observe that \( (q' t_\mathsf {l} + p' t_\mathsf {r})(q' r_\mathsf {l} + p' r_\mathsf {r}) = (q')^2 t_\mathsf {l} r_\mathsf {l} + (p')^2 t_\mathsf {r} r_\mathsf {r} \bmod p'q'\). Given that \(y_\mathsf {l}\cdot y_\mathsf {r} = g_1^{q' t_\mathsf {l}+ p' t_\mathsf {r}} \), this establishes that the combined challenge is DDH distributed. For the other case, when the challenge pair follows the random distribution, then \(Y_\mathsf {l}^{q'}\cdot Y_\mathsf {r}^{p'} = y_\mathsf {l}^{q' r_\mathsf {l}'} y_\mathsf {r}^{ p' r_\mathsf {r}'} = g_1^{ (q')^2 t_\mathsf {l} r_\mathsf {l}' + (p')^2 t_\mathsf {r} r_\mathsf {r}' } \) that can be easily seen to be uniformly distributed over \(\mathcal {X}_{n^2}\) and as a result the combined challenge is randomly distributed.

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Authors and Affiliations

  1. George Mason University, Fairfax, VA, 22030, USA

    Foteini Baldimtsi

  2. University of Edinburgh, IOG, Edinburgh, EH8 9YL, UK

    Aggelos Kiayias

  3. Hypertech Energy Labs, Athens, Greece

    Katerina Samari

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  1. Foteini Baldimtsi
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  2. Aggelos Kiayias
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  3. Katerina Samari
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Correspondence to Foteini Baldimtsi .

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Editors and Affiliations

  1. Università degli Studi di Salerno, Fisciano, Italy

    Clemente Galdi

  2. University of California at Irvine, Irvine, CA, USA

    Stanislaw Jarecki

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Baldimtsi, F., Kiayias, A., Samari, K. (2022). Watermarkable Public Key Encryption with Efficient Extraction Under Standard Assumptions. In: Galdi, C., Jarecki, S. (eds) Security and Cryptography for Networks. SCN 2022. Lecture Notes in Computer Science, vol 13409. Springer, Cham. https://doi.org/10.1007/978-3-031-14791-3_11

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  • DOI: https://doi.org/10.1007/978-3-031-14791-3_11

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