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I want to implement the homomorphic encryption scheme proposed in GSW13, and make some changes to it, but seems that there is no library that has already been implemented it. Since this is a homomorphic encryption scheme based on LWE, both Helib and SEAL have implemented some other LWE-based homomorphic encryption algorithms. I want to ask if I can use one of these two libraries to generate the public and private keys in the form required in the GSW13? And would tell me based on these two libraries, how to implement the of GSW13?

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A good resource for FHE implementations is found on Micciancio's website. Searching GSW on it points to FHEW/TFHE, although they are quite different than "standard FHE" schemes (they bootstrap after each gate in the circuit). They are implemented though.

Besides those, there are various libraries for lattice crypto (Palisade, Lol, NFLlib, all on the linked webpage) which may be useful to use in any given implementation as a starting point.

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