Fully homomorphic encryption with polylog overhead. (English) Zbl 1297.94071
Pointcheval, David (ed.) et al., Advances in cryptology – EUROCRYPT 2012. 31st annual international conference on the theory and applications of cryptographic techniques, Cambridge, UK, April 15–19, 2012. Proceedings. Berlin: Springer (ISBN 978-3-642-29010-7/pbk). Lecture Notes in Computer Science 7237, 465-482 (2012).
Summary: We show that homomorphic evaluation of (wide enough) arithmetic circuits can be accomplished with only polylogarithmic overhead. Namely, we present a construction of fully homomorphic encryption (FHE) schemes that for security parameter \(\lambda \) can evaluate any width-\(\Omega (\lambda )\) circuit with \(t\) gates in time \(t\cdot \text{polylog}(\lambda )\).
To get low overhead, we use the recent batch homomorphic evaluation techniques of Smart-Vercauteren and Brakerski-Gentry-Vaikuntanathan, who showed that homomorphic operations can be applied to “packed” ciphertexts that encrypt vectors of plaintext elements. In this work, we introduce permuting/routing techniques to move plaintext elements across these vectors efficiently. Hence, we are able to implement general arithmetic circuit in a batched fashion without ever needing to “unpack” the plaintext vectors.
We also introduce some other optimizations that can speed up homomorphic evaluation in certain cases. For example, we show how to use the Frobenius map to raise plaintext elements to powers of \(p\) at the “cost” of a linear operation.
For the entire collection see [Zbl 1239.94002].
To get low overhead, we use the recent batch homomorphic evaluation techniques of Smart-Vercauteren and Brakerski-Gentry-Vaikuntanathan, who showed that homomorphic operations can be applied to “packed” ciphertexts that encrypt vectors of plaintext elements. In this work, we introduce permuting/routing techniques to move plaintext elements across these vectors efficiently. Hence, we are able to implement general arithmetic circuit in a batched fashion without ever needing to “unpack” the plaintext vectors.
We also introduce some other optimizations that can speed up homomorphic evaluation in certain cases. For example, we show how to use the Frobenius map to raise plaintext elements to powers of \(p\) at the “cost” of a linear operation.
For the entire collection see [Zbl 1239.94002].
MSC:
| 94A60 | Cryptography |
