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Additively Homomorphic IBE from Higher Residuosity

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11442))

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Abstract

We present an identity-Based encryption (IBE) scheme that is group homomorphic for addition modulo a “large” (i.e. superpolynomial) integer, the first such group homomorphic IBE. Our first result is the construction of an IBE scheme supporting homomorphic addition modulo a poly-sized prime e. Our construction builds upon the IBE scheme of Boneh, LaVigne and Sabin (BLS). BLS relies on a hash function that maps identities to \(e^{\text {th}}\) residues. However there is no known way to securely instantiate such a function. Our construction extends BLS so that it can use a hash function that can be securely instantiated. We prove our scheme secure under the (slightly modified) \(e^{\text {th}}\) residuosity assumption in the random oracle model and show that it supports a (modular) additive homomorphism. By using multiple instances of the scheme with distinct primes and leveraging the Chinese Remainder Theorem, we can support homomorphic addition modulo a “large” (i.e. superpolynomial) integer. We also show that our scheme for \(e > 2\) is anonymous by additionally assuming the hardness of deciding solvability of a special system of multivariate polynomial equations. We provide a justification for this assumption by considering known attacks.

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Notes

  1. 1.

    LWE-based additively homomorphic IBE can be constructed with an a superpolynomial range but supporting only a theoretically bounded number of operations, albeit the bound is more than sufficient for practical purposes.

  2. 2.

    Any PPT distinguisher has only a negligible advantage (in \(\lambda \)) of distinguishing the distributions.

  3. 3.

    This is with absolute correctness. There is an alternative approach to the one we present here that achieves probabilistic correctness, but the parameters can be set so that it is correct with all but negligible probability. It is however less space efficient. The idea is that the hash function gives multiple (say \(k = \mathsf {poly}(\lambda )\)) elements whose \(e^{\text {th}}\) residue symbol is 1 and at least one of them will be an \(e^{\text {th}}\) residue with all but negligible probability. The ciphertext contains k encryptions, as opposed to \(e < k\) in our approach, thus making this approach less space-efficient than ours.

  4. 4.

    We omitted an explicit check for this in the encryption algorithm since a non-unit occurs with negligible probability.

  5. 5.

    Reported as emerging from personal communication in [34].

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Authors and Affiliations

  1. Georgetown University, Washington, D.C., USA

    Michael Clear

  2. Trinity College Dublin, Dublin, Ireland

    Ciaran McGoldrick

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  1. Michael Clear
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  2. Ciaran McGoldrick
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Correspondence to Michael Clear .

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Editors and Affiliations

  1. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. Security Research Laboratories, NEC Corporation, Kawasaki, Japan

    Kazue Sako

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© 2019 International Association for Cryptologic Research

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Clear, M., McGoldrick, C. (2019). Additively Homomorphic IBE from Higher Residuosity. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11442. Springer, Cham. https://doi.org/10.1007/978-3-030-17253-4_17

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  • DOI: https://doi.org/10.1007/978-3-030-17253-4_17

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