At CRYPTO 2018, Cascudo et al. introduced Reverse Multiplication Friendly Embeddings (RMFEs). These are a mechanism to compute $\delta$ parallel evaluations of the same arithmetic circuit over a field $F_q$ at the cost of a single evaluation of that circuit in $F_{q^d}$, where $\delta < d$. Due to this inequality, RMFEs are a useful tool when protocols require to work over $F_{q^d}$ but one is only interested in computing over $F_q$. In this work we introduce Circuit Amortization Friendly Encoding (CAFEs), which generalize RMFEs while having concrete efficiency in mind. For a Galois Ring $R = GR(2^k,d)$, CAFEs allow to compute certain circuits over $Z_{2^k}$ at the cost of a single secure multiplication in $R$. We present three CAFE instantiations, which we apply to the protocol for MPC over $Z_{2^k}$ via Galois Rings by Abspoel et al. (TCC 2019). Our protocols allow for efficient switching between the different CAFEs, as well as between computation over $R = GR(2^k,d)$ and $F_{2^{d}}$ in a way that preserves the CAFE in both rings. This adaptability leads to efficiency gains for e.g. Machine Learning applications, which can be represented as highly parallel circuits over $Z_{2^k}$ followed by bit-wise operations. From an implementation of our techniques, we estimate that an SVM can be evaluated on 250 images in parallel up to $\times 7$ more efficiently using our techniques, compared to the protocol from Abspoel et al. (TCC 2019).

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