Faster Amortized FHEW Bootstrapping Using Ring Automorphisms

Abstract

Amortized bootstrapping offers a way to simultaneously refresh many ciphertexts of a fully homomorphic encryption scheme, at a total cost comparable to that of refreshing a single ciphertext. An amortization method for FHEW-style cryptosystems was first proposed by (Micciancio and Sorrell, ICALP 2018), who showed that the amortized cost of bootstrapping n FHEW-style ciphertexts can be reduced from $\tilde{O}(n)$ basic cryptographic operations to just $\tilde{O}(n^{\epsilon })$, for any constant $\epsilon >0$. However, despite the promising asymptotic saving, the algorithm was rather impractical due to a large constant (exponential in $1/\epsilon$) hidden in the asymptotic notation. In this work, we propose an alternative amortized bootstrapping method with much smaller overhead, still achieving $O(n^\epsilon )$ asymptotic amortized cost, but with a hidden constant that is only linear in $1/\epsilon$, and with reduced noise growth. This is achieved following the general strategy of (Micciancio and Sorrell), but replacing their use of the Nussbaumer transform, with a much more practical Number Theoretic Transform, with multiplication by twiddle factors implemented using ring automorphisms. A key technical ingredient to do this is a new “scheme switching” technique proposed in this paper which may be of independent interest.