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Bylines, paper authorship, correspondence addresses and the founder blurb under each post named an individual. They name the organisation that published the work instead: the brand on posts and pages, the research byline on papers, and research@ in place of a personal address. Both are what these repositories already used elsewhere — nothing new was coined. Co-authors are untouched. Where a paper named someone else alongside, they stay named; only the one byline changed. Bibliography entries, running heads, PDF metadata and affiliations move with it, so a citation still resolves to whoever the cited paper names. Sample and fixture data uses the placeholder name these tests already use elsewhere, so no assertion changed meaning.
79 lines
3.2 KiB
TeX
79 lines
3.2 KiB
TeX
\documentclass[11pt,a4paper]{article}
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\usepackage{../shared/luxcover}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amssymb,amsfonts,amsthm}
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\usepackage{graphicx}
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\usepackage{hyperref}
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\usepackage{algorithm}
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\usepackage{algpseudocode}
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\usepackage{xcolor}
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\usepackage{booktabs}
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\usepackage{enumitem}
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\usepackage{tikz}
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\hypersetup{colorlinks=true,linkcolor=black,citecolor=blue,urlcolor=blue}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{corollary}[theorem]{Corollary}
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{definition}{Definition}[section]
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\newtheorem{remark}{Remark}[section]
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\title{\textbf{Torus Threshold Fully Homomorphic Encryption:\\Boolean Circuits on Encrypted Data\\with Distributed Decryption}}
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\author{
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Lux Research\thanks{Corresponding author: [email protected]}\\
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\textit{Lux Industries, Inc.}\\
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\texttt{[email protected]}
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}
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\date{2025}
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\begin{document}
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\luxcoverpage
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\begin{abstract}
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We present a threshold variant of Torus Fully Homomorphic Encryption (TFHE) that enables
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evaluation of arbitrary boolean circuits on encrypted bits with distributed decryption
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across $t$-of-$n$ parties. The scheme encrypts individual bits as Learning With Errors
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(LWE) ciphertexts over the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, performs gate
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evaluation through programmable bootstrapping (blind rotation), and requires no interaction
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between parties during computation---only the decryption phase requires threshold
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cooperation.
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We describe the core TFHE gate evaluation pipeline: (1) LWE encryption of a single bit
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with noise parameter $\sigma$, (2) blind rotation using RGSW ciphertexts of the secret
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key bits to evaluate a lookup table encoding the gate function, (3) sample extraction
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and modulus switching to produce a fresh LWE ciphertext of the gate output, and
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(4) the CMPCOMBINE optimization that fuses comparison and bootstrap operations to evaluate
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multi-bit comparisons in a single bootstrapping pass.
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We formalize the threshold decryption protocol: each party holds a share of the LWE
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secret key $\mathbf{s}$, computes a partial decryption $d_i = \langle \mathbf{c}, \mathbf{s}_i \rangle$,
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and the combiner reconstructs the plaintext from $t$ partial decryptions. We prove that
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threshold TFHE achieves IND-CPA security under the LWE assumption with the same parameters
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as standard TFHE (N=1024, $q \approx 2^{27}$, providing 128-bit security).
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The implementation builds on the \texttt{luxfi/lattice} library for ring operations
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and supports all 9 boolean gates (AND, OR, XOR, NOT, NAND, NOR, XNOR, MUX, MAJORITY).
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Benchmarks show 12ms per bootstrapped gate on a single core, enabling EVM-level boolean
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circuit evaluation on encrypted data with practical performance.
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\end{abstract}
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\tableofcontents
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\newpage
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%% -----------------------------------------------------------------------
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%% 1. INTRODUCTION
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%% -----------------------------------------------------------------------
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\input{sections/01-introduction}
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\input{sections/02-tfhe-preliminaries}
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\input{sections/03-boolean-gate-evaluation}
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\input{sections/04-threshold-decryption}
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\input{sections/05-parameter-selection}
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\input{sections/06-fhe-vm-integration}
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\input{sections/07-empirical-evaluation}
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\input{sections/99-conclusion}
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\end{document}
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