Aug 2026· IACR Communications in Cryptology· 0 citations· 19 references
TL;DR
Experimental evaluations demonstrate that encrypted low-rank matrix multiplications achieve both significant runtime improvements and reduction of ciphertext sizes over direct or tree-based encrypted multiplications while maintaining the prescribed accuracy.
Abstract
Privacy-preserving machine learning and encrypted statistics increasingly require evaluating long chains of matrix products directly on ciphertexts. In the CKKS homomorphic encryption scheme, however, every multiplication amplifies noise and enlarges ciphertexts, so the available precision budget is exhausted after only a few products. This work establishes
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(compressed FHE), a unified analytical and empirical framework that integrates low-rank matrix factorization techniques into the CKKS homomorphic encryption scheme. Its central idea is a co-design: instead of tuning the low-rank approximation and the cryptographic parameters in isolation,
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balances the two error sources against each other so that neither wastes precision. Theoretical bounds are derived for the accumulation of relative error across sequences of factorized matrices, leading to an explicit expression for the attainable computation depth as a function of target accuracy, norm amplification behavior, and per-layer approximation quality. Extensions to tree-based evaluation structures are also formulated, allowing depth to scale logarithmically with the number of factors.
Concretely, this co-design is realized as a precision-balancing model that, for a target accuracy, automatically selects the CKKS parameters: the polynomial modulus degree, the modulus chain, and the scaling factor.
Experimental evaluations demonstrate that encrypted low-rank matrix multiplications achieve both significant runtime improvements and reduction of ciphertext sizes over direct or tree-based encrypted multiplications while maintaining the prescribed accuracy.
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is agnostic to other CKKS optimizations and can be combined with them for further gains.
We present Clifford fully homomorphic encryption (FHE), the first ring-learning with errors (RLWE)-based FHE scheme with native support for all Clifford algebra operations, enabling privacy-preserving computation on geometric data. Unlike conventional FHE schemes that flatten geometric structure into scalar operations, our approach maintains the algebraic properties of multi-vectors through homomorphic geometric products. Building on this foundation, we introduce geometric neural networks that operate directly on encrypted multi-vectors, achieving the first-ever demonstration of privacy-preserving geometric deep learning. Our construction encrypts three-dimensional point clouds as Cl(3,0) multi-vectors and performs encrypted classification with less than 1% accuracy loss compared to plaintext, completing inference in under 60 s. These results demonstrate that geometric algebra (GA) provides unique advantages for both cryptographic constructions (enabling structure-preserving encryption) and machine learning (ML) (enabling privacy-preserving geometric learning), opening new pathways at the intersection of cryptography, ML and applied mathematics. This article is part of the theme issue 'Modern applications of geometric algebra'.
D. Silva· Philosophical transactions....· 2 citations
Fully homomorphic encryption (FHE) allows a server to run a language model directly on encrypted user prompts, but current approaches remain prohibitively slow. Ciphertexts natively support only addition, multiplication, and rotation, and multiplications may be composed only to a bounded depth before a costly bootstrapping operation is needed to continue. Every nonlinearity must therefore be approximated by an iterative method, and each iteration uses multiplications. A higher iteration count buys precision but exhausts the available depth faster and triggers more bootstraps, which dominate latency. Existing approaches fix the iteration counts uniformly across the model rather than tailoring them to each site's error tolerance. We introduce Homomorphic Encryption-Aware Training (HEAT), a fine-tuning method that makes the per-nonlinearity iteration counts learnable, enabling them and the model weights to co-adapt during training. HEAT optimizes iterations with respect to the task objective, allowing the model to adapt to approximation errors encountered during inference without architectural changes or retraining from scratch. On encrypted GPT-2 decoding, HEAT reduces iterations by $3.1\times$, bootstraps by $1.6\times$, and end-to-end latency by $1.4\times$, while improving decode agreement over the calibrated baseline.
Alessandro Zirilli, Davide Marincione, Evgenios M. Kornaropoulos et al.· 0 citations
Field trace evaluation has emerged as a powerful tool in fully homomorphic encryption, with broad applications ranging from bootstrapping algorithms to privacy-preserving protocols. Recent advances have significantly reduced its noise growth by combining tower-based evaluation strategies with rescaling operations. However, existing analyses rely on uniform noise bounds that fail to capture the actual noise behavior across different coefficients, leading to substantial gaps between theoretical estimates and empirical observations.
In this work, we present a refined algebraic analysis of trace evaluation over power-of-two cyclotomics that uncovers structured cancellation effects among noise coefficients induced by subsequent linear operators, in particular the trace mappings of subextensions. We show that, except for the constant term, the variance of each output noise coefficient depends on the 2-adic valuation of its index, yielding bounds that improve upon prior uniform estimates by a factor of
O
(
log
n
)
both for non-constant coefficients and after a subsequent plaintext-ciphertext multiplication, where
n
is the ring degree. We further extend our analysis to two typical algorithmic applications of trace evaluation. For ciphertext packing, we derive a non-recursive formulation that admits a cleaner structure and slightly tighter noise estimates. For coefficient extraction, our coefficient-wise analysis improves upon prior uniform variance bounds by factors ranging from
Θ
(
n
)
to
Θ
(
n
2
)
for non-constant coefficients and by a factor of
O
(
n
)
after post-multiplication. Experimental results confirm that the observed noise variances follow the coefficient-wise pattern predicted by our analysis and demonstrate pronounced improvements over existing estimates, providing effective guidance for parameter selection and system configuration in practice.
Han Xia· IACR Communications in Crypt...· 0 citations
This work proposes a practical non-interactive encrypted retrieval framework for RAG based on threshold selection, and introduces a precision-stable mask polarization method that ensures accurate recovery of selected documents.
Yang Gao, Gang Quan, Scott Piersall et al.· arXiv.org· 0 citations
This work proposes a watermarking technique for RLWE-based HE ciphertexts by exploiting the algebraic structure of RLWE polynomials and introduces two practical schemes that preserve the original security of HE while maintaining correctness and watermark robustness.
This work presents a framework that reformulates HE-aware model design as a constrained neural architecture search problem, where the objective is to identify architectures that are both cryptographically feasible and computationally efficient while preserving task performance.
Reeshav Chowdhury, Anoop Mishra, Deepak Khazanchi et al.· ACM Transactions on Internet...· 0 citations
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