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As it is discussed here that the highest security any homomorphic encryption scheme is at most IND-CCA1, Is there any known fully homomorphic encryption scheme that achieves this security level? Out of many schemes mentioned here (or otherwise) which of them are IND-CCA1 secure ? (not necessarily practical though)

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sashank
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A recent paper: https://arxiv.org/abs/1804.06862 claims to be fully homomorphic with IND-CCA1. However it is a symmetric encryption scheme and I'm not sure if IND-CCA1 is an appropriate measure in this setting.

Besides this, I think security is based on the hardness of solving overdetermined systems of quadratic equation over $\mathbb{Z}_{(p\cdot q)^2)}$ rings, which then might not be Quantum-hard. If I'm not mistaken, there are sixteen quadratic equations and if we can factor $(p\cdot q)^2$ then these equations can be efficiently solved.

Otherwise the scheme is pretty efficient, in terms of key-length, decryption and encryption time, I think. It is noise free and based on ring-automorphisms for matrices with elements from the quaternion-ring over some $\mathbb{Z}_{(p\cdot q)^2)}$ ring.

However as the paper appeared only recently, an in-deep analysis, besides the argumentation auf the authors, is not yet availablabe as far as I know.

Mark Neuhaus
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