International Association
for Cryptologic Research

Fully Homomorphic Encryption (FHE) is an encryption scheme that allows for computation to be performed directly on encrypted data. FHE effectively closes the loop on secure and outsourced computing; data is encrypted not only during rest and transit, but also during processing. Moreover, modern FHE schemes such as RNS-CKKS (with the canonical slot encoding) encrypt one-dimensional floating-point vectors, which makes such a scheme an ideal candidate for building private machine learning systems. However, RNS-CKKS provides a limited instruction set: SIMD addition, SIMD multiplication, and cyclic rotation of these encrypted vectors. This restriction makes performing multi-dimensional tensor operations (such as those used in machine learning) challenging. Practitioners must pack multi-dimensional tensors into 1-D vectors and map tensor operations onto this one-dimensional layout rather than their traditional nested structure. And while prior systems have made significant strides in automating this process, they often hide critical packing decisions behind layers of abstraction, making debugging, optimizing, and building on top of these systems difficult.

In this work we ask: can we build an FHE tensor system with a straightforward and transparent packing strategy regardless of the tensor operation? We answer affirmatively and develop a packing strategy based on Einstein summation (einsum) notation. We find einsum notation to be ideal for our approach since the notation itself explicitly encodes the dimensional structure and operation directly into its syntax, naturally exposing how tensors should be packed and manipulated in FHE. We make use of einsum's explicit language to decompose einsum expressions into a fixed set of FHE-friendly operations: dimension expanding and broadcasting, element-wise multiplication, and a reduction along the contraction dimensions.

We implement our design and present EinHops, which stands for Einsum Notation for Homomorphic Tensor Operations. EinHops is a minimalist system that factors einsum expressions into a fixed sequence of FHE operations, enabling developers to perform complex tensor operations using RNS-CKKS while maintaining full visibility into the underlying packing strategy. We evaluate EinHops on a range of tensor operations from a simple transpose to complex multi-dimensional contractions. We show that the explicit nature of einsum notation allows us to build an FHE tensor system that is simple, general, and interpretable. We open-source EinHops at the following repository: https://github.com/baahl-nyu/einhops.