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Optimal Broadcast Encryption and CP-ABE from Evasive Lattice Assumptions

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Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13276))

Abstract

We present a new, simple candidate broadcast encryption scheme for N users with parameter size \(\textsf {poly}(\log N)\). We prove security of our scheme under a non-standard variant of the LWE assumption where the distinguisher additionally receives short Gaussian pre-images while avoiding zeroizing attacks. This yields the first candidate optimal broadcast encryption that is plausibly post-quantum secure, and enjoys a security reduction to a simple assumption. As a secondary contribution, we present a candidate ciphertext-policy attribute-based encryption (CP-ABE) scheme for circuits of a-priori bounded polynomial depth where the parameter size is independent of the circuit size, and prove security under an additional non-standard assumption.

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Notes

  1. 1.

    For simplicity of exposition and due to the sheer complexity and impracticality of the ensuing schemes, we ignore obfuscation-based broadcast in the rest of the introduction, deferring a comparison to Sect. 2.3.

  2. 2.

    Note that the error distribution \(\mathbf {e}\cdot \mathbf {B}^{-1}(\mathbf {P})\) in \(\mathbf {c}^*\) is different from the fresh Gaussian error \(\mathbf {e}''\). Differences in error distributions can make or break a scheme if \(\mathbf {c}^*\) has small norm, but we do not know attacks exploiting these differences when \(\mathbf {c}^*\) has large norm, as is the case here.

  3. 3.

    As a point of comparison, we have examples such as k-LWE [38] and inner product functional encryption [3] based on LWE where it is easy to obtain a few such equations, but the equations do not information-theoretically determine the secret values.

  4. 4.

    Security based on evasive LWE can be viewed as ruling out restricted adversaries that replaces \(\mathbf {s}\mathbf {B}+\mathbf {e}, \mathbf {B}^{-1}(\mathbf {P})\) with their product \(\mathbf {s}\mathbf {P}+\mathbf {e}''\) (with fresh noise) and ignoring \(\mathbf {B}^{-1}(\mathbf {P})\) thereafter. Viewed this way, evasive LWE can be seen as a partial analogue of the generic/algebraic group model used in group and pairing-based cryptography. Several works studied analogues of the generic group model for multi-linear maps [9, 30], but they were in the zeroizing regime.

  5. 5.

    That is, \(x_i+x_j\) corresponds to \(\mathbf {A}_i + \mathbf {A}_j\) and \(x_i \cdot x_j\) corresponds to \(\mathbf {A}_i \cdot \mathbf {A}_j\) instead of \(\mathbf {A}_i \cdot \mathbf {G}^{-1}(\mathbf {A}_j)\). More generally, we can represent a circuit f of depth d and size s as a polynomial comprising the sum of s monomials, each of total degree at most \(2^d\). Then, \(\mathbf {A}_f = f(\mathbf {A}_1,\ldots ,\mathbf {A}_\ell )\).

  6. 6.

    As explained in [12], “To support multiplication and addition of constants, we may assume that we have an extra 0-th input to the circuit that always carries the value 1.” That is, we will set \(\ell = \lceil \log N \rceil +1\) in our CP-ABE scheme.

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  1. NTT Research and ENS, Paris, France

    Hoeteck Wee

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  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Wee, H. (2022). Optimal Broadcast Encryption and CP-ABE from Evasive Lattice Assumptions. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_8

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