EinHops: Einsum Notation for Expressive Homomorphic Operations on RNS-CKKS Tensors

TL;DR

EinHops introduces a novel approach to perform multi-dimensional tensor operations in Fully Homomorphic Encryption (FHE) using Einstein summation notation, improving transparency and flexibility in encrypted tensor computations.

Contribution

The paper presents EinHops, a system that decomposes einsum expressions into FHE-friendly operations, enabling explicit, interpretable, and general encrypted tensor processing.

Findings

Successfully performs complex tensor operations like transpose and contraction in FHE.

Provides full visibility into tensor packing strategies during encrypted computation.

Demonstrates simplicity and generality of the einsum-based approach in FHE tensor operations.

Abstract

Fully Homomorphic Encryption (FHE) is an encryption scheme that allows for computation to be performed directly on encrypted data, effectively closing the loop on secure and outsourced computing. Data is encrypted not only during rest and transit, but also during processing. However, FHE provides a limited instruction set: SIMD addition, SIMD multiplication, and cyclic rotation of 1-D vectors. This restriction makes performing multi-dimensional tensor operations challenging. Practitioners must pack these 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 approach multi-dimensional tensor operations in FHE through Einstein summation (einsum) notation. Einsum notation explicitly encodes dimensional structure and operations in its syntax, naturally exposing how tensors should be packed and transformed. We decompose einsum expressions into a fixed set of FHE-friendly operations. We implement our design and present EinHops, a minimalist system that factors einsum expressions into a fixed sequence of FHE operations. EinHops enables developers to perform encrypted tensor operations using FHE 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.