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Title:
Error Correction and Ciphertext Quantization in Lattice Cryptography
Authors: Daniele Micciancio, Mark Schultz
Abstract:Recent work in the design of rate 1 - o(1) lattice-based cryptosystems have used two distinct design paradigms, namely replacing the noise-tolerant encoding m \mapsto (q/2)m present in many lattice-based cryptosystems with a more efficient encoding, and post-processing traditional lattice-based ciphertexts with a lossy compression algorithm, using a technique very similar to the technique of vector quantization'' within coding theory. We introduce a framework for the design of lattice-based encryption that captures both of these paradigms, and prove information-theoretic rate bounds within this framework. These bounds separate the settings of trivial and non-trivial quantization, and show the impossibility of rate $1 - o(1)$ encryption using both trivial quantization and polynomial modulus. They furthermore put strong limits on the rate of constructions that utilize lattices built by tensoring a lattice of small dimension with $\mathbb{Z}^k$, which is ubiquitous in the literature. We additionally introduce a new cryptosystem, that matches the rate of the highest-rate currently known scheme, while encoding messages with a gadget’', which may be useful for constructions of Fully Homomorphic Encryption.
ePrint: https://eprint.iacr.org/2023/525
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