VLWE: Variety-based Learning with Errors for Vector Encryption through Algebraic Geometry
Dongfang Zhao
TL;DR
This work introduces Variety-LWE (VLWE), a novel lattice-hard problem defined over multivariate polynomial rings constrained by algebraic varieties with no mixed terms, enabling coordinate-wise computations and direct vector encryption. The authors establish worst-case to average-case hardness by reducing VLWE to multiple independent Ideal-SVP instances and analyze resilience to classical and quantum attacks, including hybrid algebraic-lattice strategies. They show VLWE offers more controllable error propagation under vectorized homomorphic operations and provide a VLWE-based vector encryption scheme with added relinearization to manage noise growth. Compared to Ring-LWE and Module-LWE, VLWE increases lattice dimension and key sizes but benefits from natural suitability for high-dimensional, structured data tasks such as privacy-preserving machine learning and encrypted vector processing. The work further discusses parameter selection, security implications, and practical considerations, framing VLWE as a distinct, independent paradigm in lattice cryptography leveraging algebraic geometry to enable new cryptographic capabilities beyond traditional polynomial quotient rings.
Abstract
Lattice-based cryptography is a foundation for post-quantum security, with the Learning with Errors (LWE) problem as a core component in key exchange, encryption, and homomorphic computation. Structured variants like Ring-LWE (RLWE) and Module-LWE (MLWE) improve efficiency using polynomial rings but remain constrained by traditional polynomial multiplication rules, limiting their ability to handle structured vectorized data. This work introduces Variety-LWE (VLWE), a new structured lattice problem based on algebraic geometry. Unlike RLWE and MLWE, which use polynomial quotient rings with standard multiplication, VLWE operates over multivariate polynomial rings defined by algebraic varieties. A key difference is that these polynomials lack mixed variables, and multiplication is coordinate-wise rather than following standard polynomial multiplication. This enables direct encoding and homomorphic processing of high-dimensional data while preserving worst-case to average-case hardness reductions. We prove VLWE's security by reducing it to multiple independent Ideal-SVP instances, demonstrating resilience against classical and quantum attacks. Additionally, we analyze hybrid algebraic-lattice attacks, showing that existing Grobner basis and lattice reduction methods do not directly threaten VLWE. We further construct a vector homomorphic encryption scheme based on VLWE, supporting structured computations while controlling noise growth. This scheme offers advantages in privacy-preserving machine learning, encrypted search, and secure computations over structured data. VLWE emerges as a novel and independent paradigm in lattice-based cryptography, leveraging algebraic geometry to enable new cryptographic capabilities beyond traditional polynomial quotient rings.