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

2025-07-10

AI Summary

Fully homomorphic encryption (FHE) natively supports only one-dimensional SIMD addition, multiplication, and cyclic rotations, making efficient multidimensional tensor operations challenging. Existing FHE systems rely on implicit tensor flattening and manual mapping, resulting in opaque packing strategies, poor debuggability, and limited scalability. Method: This paper introduces Einstein summation (einsum) notation to FHE for the first time, explicitly modeling tensor structure and operation semantics, and automatically decomposing einsum expressions into FHE-compatible primitive operators. Built upon the RNS-CKKS scheme, our system integrates einsum parsing, dimensionality-reduction mapping, optimized cyclic rotations, and SIMD vector operations to support general multidimensional operations—including transposition and contraction. Contribution/Results: The lightweight framework significantly improves interpretability, generality, and debuggability without compromising performance. Experimental evaluation validates its effectiveness, and the implementation is open-sourced.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Learning & Optimization for CV

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• Enables multi-dimensional tensor operations in FHE
• Simplifies encrypted tensor packing and transformation
• Improves debugging and optimization of FHE systems

Innovation

• Uses einsum notation for tensor operations
• Decomposes einsum into FHE-friendly operations
• Provides full visibility into packing strategy