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Non-zero Inner Product Encryption Schemes from Various Assumptions: LWE, DDH and DCR

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Public-Key Cryptography – PKC 2019 (PKC 2019)

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Abstract

In non-zero inner product encryption (NIPE) schemes, ciphertexts and secret keys are associated with vectors and decryption is possible whenever the inner product of these vectors does not equal zero. So far, much effort on constructing bilinear map-based NIPE schemes have been made and this has lead to many efficient schemes. However, the constructions of NIPE schemes without bilinear maps are much less investigated. The only known other NIPE constructions are based on lattices, however, they are all highly inefficient due to the need of converting inner product operations into circuits or branching programs.

To remedy our rather poor understanding regarding NIPE schemes without bilinear maps, we provide two methods for constructing NIPE schemes: a direct construction from lattices and a generic construction from schemes for inner products (LinFE). For our first direct construction, it highly departs from the traditional lattice-based constructions and we rely heavily on new tools concerning Gaussian measures over multi-dimensional lattices to prove security. For our second generic construction, using the recent constructions of LinFE schemes as building blocks, we obtain the first NIPE constructions based on the DDH and DCR assumptions. In particular, we obtain the first NIPE schemes without bilinear maps or lattices.

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Notes

  1. 1.

    We note that Goyal et al. [GPSW06] propose an ABE scheme for \(\mathbf {NC}^1\) circuit, which in turn implies a NIPE scheme, since the computation of inner products can be performed in \(\mathbf {NC}^1\). However, the resulting construction is highly inefficient.

  2. 2.

    The term LinFE is borrowed from [ALS16]. It is named as such, since it is a special type of functional encryption scheme restricted to the class of linear functions.

  3. 3.

    IPE is a special kind of ABE where decryption is possible iff the inner product of the vectors corresponding to a ciphertext and a private key is 0. This should not be confused with LinFE, where the decryption is always possible and the decryption result is the inner product itself.

  4. 4.

    Note that for the case \(q = p^k\) for some \(k \in \mathbb N\), we set the statistical distance to be \(n^{-\omega (1)}\) rather than \(2^{-{\varOmega }(n)}\) as in [ALS16], Lem. 9.

  5. 5.

    Although \(\mathbf {h}^{(j')} \in \mathbb Z_p^{\ell }\) and \(\mathbf {\mathsf{h}}^{(j')} \in \mathbb Z^\ell \) are in some sense identical, we intentionally write it redundantly in this form for consistency with the other predicate vectors \(\mathbf {y}\), i.e., \((\mathbf {\mathsf{h}}^{(j')}, \mathsf{sk}_{\mathbf {h}^{(j')}})\) acts as a valid secret key for the predicate vector \(\mathbf {h}^{(j')}\).

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Acknowledgement

The first author was partially supported by JST CREST Grant Number JPMJCR1302 and JSPS KAKENHI Grant Number 17J05603. The second author was supported by JST CREST Grant Number JPMJCR1688 and JSPS KAKENHI Grant Number 16K16068.

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Correspondence to Shuichi Katsumata .

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Katsumata, S., Yamada, S. (2019). Non-zero Inner Product Encryption Schemes from Various Assumptions: LWE, DDH and DCR. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham. https://doi.org/10.1007/978-3-030-17259-6_6

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