Damien Stehlé

Youngjin Bae, Jung Hee Cheon, Guillaume Hanrot et al.

Homomorphic encryption is a cryptographic paradigm allowing to compute on encrypted data, opening a wide range of applications in privacy-preserving data manipulation, notably in AI. Many of those applications require significant linear algebra computations (matrix-vector products, and matrix-matrix products). This central role of linear algebra computations goes far beyond homomorphic algebra and applies to most areas of scientific computing. This high versatility led, over time, to the development of a set of highly optimized routines, specified in 1979 under the name BLAS (basic linear algebra subroutines). Motivated both by the applicative importance of homomorphic linear algebra and the access to highly efficient implementations of cleartext linear algebra able to draw the most out of available hardware, we explore the connections between CKKS-based homomorphic linear algebra and floating-point plaintext linear algebra. The CKKS homomorphic encryption system is the most natural choice in this setting, as it natively handles real numbers and offers a large SIMD parallelism. We provide reductions for matrix-vector products, vector-vector products for moderate-sized to large matrices to their plaintext equivalents. Combined with BLAS, we demonstrate that the efficiency loss between CKKS-based encrypted square matrix multiplication and double-precision floating-point square matrix multiplication is a mere 4-12 factor, depending on the precise situation.

Jaiyoung Park, Sejin Park, Jai Hyun Park et al.

As large language models (LLMs) become ubiquitous, privacy concerns pertaining to inference keep growing. Fully homomorphic encryption (FHE) has emerged as a primary cryptographic solution for non-interactive confidential LLM inference. However, existing solutions scale poorly with input token length, focusing on small models or input sizes. They also suffer from large outlier values, which strongly impact the evaluation of non-linear layers, leading to heavy polynomial approximation costs. We scale up FHE-based LLM inference in two directions. First, we accelerate FHE-based inference for 128 encrypted tokens. We adopt ML techniques (token prepending and orthogonal rotations) to mitigate outlier impacts on the FHE evaluation of non-linear layers. Separately, we devise a novel polynomial evaluation method for sparsely-packed ciphertexts to speed up our homomorphic SoftMax implementation. We combine these with recent fast homomorphic linear algebra techniques, achieving significantly improved efficiency. Second, we expand the prompt size up to thousands of tokens for contexts where only the final part of the input is sensitive and encrypted. Processing this requires handling standard plaintext-plaintext and ciphertext-ciphertext components, alongside a wide homomorphic computation for a novel plaintext-ciphertext component. To address this, we devise a dedicated homomorphic linear algebra algorithm, building a shallow homomorphic attention circuit that minimizes bootstrapping costs. Based on these ingredients, we present a CKKS-based end-to-end implementation of Llama-3-8B private inference. On 8 NVIDIA RTX PRO 6000 GPUs, 128 encrypted tokens take 20s for summarization and 18s/token for generation (vastly outperforming the SOTA 295s on costlier H100 GPUs). For a heterogeneous 4096-token input (last 128 encrypted), it takes 64s for summarization and 22s/token for generation.

Jincheol Ha, Guillaume Hanrot, Taeyeong Noh et al.

Among biometric verification systems, irises stand out because they offer high accuracy even in large-scale databases. For example, the World ID project aims to provide authentication to all humans via iris recognition, with millions already registered. Storing such biometric data raises privacy concerns, which can be addressed using privacy-enhancing techniques. Bloemen et al. describe a solution based on 2-out-of-3 Secret-Sharing Multiparty Computation (SS-MPC), for the World ID setup. In terms of security, unless an adversary corrupts 2~servers, the iris codes remain confidential and nothing leaks beyond the result of the computation. Their solution is able to match~$32$ users against a database of~$2^{22}$ iris codes in~$\approx 2$s , using~24 H100 GPUs, more than 40~communication rounds and $81$GB/party of data transferred (the timing assumes a network speed above~3Tb/s). In the present work, we explore the use of Threshold Fully Homomorphic Encryption (ThFHE) for the same task. The ThFHE solution brings a number of security advantages: no trusted setup, the encrypted database and queries can be public, the secret can be distributed among many parties, and active security can be added without significant performance degradation. Our proof-of-concept implementation of the computation phase handles $32$~eyes against a database of $7\cdot 2^{14}$ iris codes in~$\approx 1.8$s ($\approx 0.33s$ for 4 eyes against the same database), using 8 RTX-5090 GPUs. To this, one should add~2 to 3 rounds of communication (depending on deployment choice). We perform the matching using the CKKS (Th)FHE scheme. Our main technical ingredients are the use of recent progress on FHE-based linear algebra boosted using int8 GPU operations, and the introduction of a technique reducing the number of ciphertexts to be processed as early as possible.

Zvika Brakerski, Elena Kirshanova, Damien Stehlé et al.

The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography. In particular, it is one of the most prominent candidates for secure post-quantum cryptography. Understanding its quantum complexity is therefore an important goal. We show that under quantum polynomial time reductions, LWE is equivalent to a relaxed version of the dihedral coset problem (DCP), which we call extrapolated DCP (eDCP). The extent of extrapolation varies with the LWE noise rate. By considering different extents of extrapolation, our result generalizes Regev's famous proof that if DCP is in BQP (quantum poly-time) then so is LWE (FOCS'02). We also discuss a connection between eDCP and Childs and Van Dam's algorithm for generalized hidden shift problems (SODA'07). Our result implies that a BQP solution for LWE might not require the full power of solving DCP, but rather only a solution for its relaxed version, eDCP, which could be easier.

Zvika Brakerski, Adeline Langlois, Chris Peikert et al.

We show that the Learning with Errors (LWE) problem is classically at least as hard as standard worst-case lattice problems, even with polynomial modulus. Previously this was only known under quantum reductions. Our techniques capture the tradeoff between the dimension and the modulus of LWE instances, leading to a much better understanding of the landscape of the problem. The proof is inspired by techniques from several recent cryptographic constructions, most notably fully homomorphic encryption schemes.