Two-limb CRT Ring-LWE encryption with exact decryption and public re-randomization

Abstract

Anonymity infrastructures such as mix networks, anonymous storage, and privacy-preserving replication rely on public re-randomization: any party holding only public information can transform a ciphertext into a fresh-looking encryption of the same plaintext, hiding the linkage between the two. Classical ElGamal-based solutions are broken by quantum adversaries, while existing lattice-based alternatives carry very large ciphertexts with unanalyzed noise growth, rely on heavyweight homomorphic-encryption stacks with approximate (rounded) decryption, or lack a precise analysis of how many re-randomizations are safe. We address this gap with a practical Ring Learning with Errors (Ring-LWE) public-key encryption scheme supporting public re-randomization without ciphertext growth. Our construction is Lyubashevsky–Peikert–Regev/Fan–Vercauteren (LPR/BFV)-style encryption over R=Z[x]/(xn+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\mathbb {Z}[x]/(x^n+1)$$\end{document} with n=4096\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=4096$$\end{document}, engineered around a two-limb Chinese Remainder Theorem (CRT) modulusq=t·q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=t\cdot q_2$$\end{document} with 32-bit primes. Embedding plaintext as ΔM=q2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta M = q_2 M$$\end{document} makes the message vanish modulo q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_2$$\end{document}, so the q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_2$$\end{document}-limb carries only the decryption noise, enabling exact message recovery without rounding. We prove correctness with explicit decryption-failure bounds that remain valid under repeated re-randomization, via an aggregation lemma showing that arbitrarily many re-randomizations affect decryption only through a single aggregated randomness triple. We also prove that two-limb ciphertexts are pseudorandom (indistinguishable from uniform, IND$) under Decision Ring-LWE over the combined modulus q=tq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=tq_2$$\end{document}; security against chosen-plaintext attack (IND-CPA) and re-randomization unlinkability follow. A constant-time Rust implementation encrypts in 0.80 ms, re-randomizes in 0.51 ms, and decrypts in 0.21 ms per 64 KiB ciphertext carrying 15.5 KiB of payload on a fixed-frequency 3.8 GHz CPU—on par with a modulus-matched Microsoft SEAL baseline—and passes timing-leakage tests. Empirical noise simulations validate the analysis.