## CryptoDB

### Noah Stephens-Davidowitz

#### Affiliation: Princeton University

#### Publications

**Year**

**Venue**

**Title**

2020

CRYPTO

Slide Reduction, Revisited—Filling the Gaps in SVP Approximation
📺
Abstract

We show how to generalize Gama and Nguyen's slide reduction algorithm [STOC '08] for solving the approximate Shortest Vector Problem over lattices (SVP) to allow for arbitrary block sizes, rather than just block sizes that divide the rank n of the lattice. This leads to significantly better running times for most approximation factors. We accomplish this by combining slide reduction with the DBKZ algorithm of Micciancio and Walter [Eurocrypt '16].
We also show a different algorithm that works when the block size is quite large---at least half the total rank. This yields the first non-trivial algorithm for sublinear approximation factors.
Together with some additional optimizations, these results yield significantly faster provably correct algorithms for \delta-approximate SVP for all approximation factors n^{1/2+\eps} \leq \delta \leq n^{O(1)}, which is the regime most relevant for cryptography. For the specific values of \delta = n^{1-\eps} and \delta = n^{2-\eps}, we improve the exponent in the running time by a factor of 2 and a factor of 1.5 respectively.

2020

CRYPTO

Lattice Reduction for Modules, or How to Reduce ModuleSVP to ModuleSVP
📺
Abstract

We show how to generalize lattice reduction algorithms to module lattices. Specifically, we reduce $\gamma$-approximate ModuleSVP over module lattices with rank $k \geq2$ to $\gamma'$-approximate ModuleSVP over module lattices with rank $2 \leq \beta \leq k$. To do so, we modify the celebrated slide-reduction algorithm of Gama and Nguyen to work with module filtrations, a high-dimensional generalization of the ($\Z$-)basis of a lattice.
The particular value of $\gamma$ that we achieve depends on the underlying number field $K$, the order $R \subseteq \mathcal{O}_K$, and the embedding (as well as, of course, $k$ and $\beta$). However, for reasonable choices of these parameters, the approximation factor that we achieve is surprisingly close to the one achieved by ``plain'' lattice reduction algorithms, which require an arbitrary SVP oracle in the same dimension. In other words, we show that ModuleSVP oracles are nearly as useful as SVP oracles for solving approximate ModuleSVP in higher dimensions.
Our result generalizes the recent independent result of Lee, Pellet-Mary, Stehl\'e, and Wallet, which works in the important special case when $\beta = 2$ and $R = \mathcal{O}_K$ is the ring of integers of $K$ under the canonical embedding. Indeed, at a high level our reduction can be thought of as a generalization of theirs in roughly the same way that block reduction generalizes LLL reduction.

2018

PKC

New (and Old) Proof Systems for Lattice Problems
Abstract

We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem $$\textsf {GapSPP}$$GapSPP of approximating the $$\varepsilon $$ε-smoothing parameter (for some $$\varepsilon < 1/2$$ε<1/2) of an n-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and $$\textsf {GapSPP}$$GapSPP has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that $$\textsf {GapSPP}$$GapSPP admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly $$\sqrt{n}$$n. Specifically:There is a noninteractive SZK proof for $$O(\log (n) \sqrt{\log (1/\varepsilon )})$$O(log(n)log(1/ε))-approximate $$\textsf {GapSPP}$$GapSPP. Moreover, for any negligible $$\varepsilon $$ε and a larger approximation factor $$\widetilde{O}(\sqrt{n \log (1/\varepsilon )})$$O~(nlog(1/ε)), there is such a proof with an efficient prover.There is an (interactive) SZK proof with an efficient prover for $$O(\log n + \sqrt{\log (1/\varepsilon )/\log n})$$O(logn+log(1/ε)/logn)-approximate coGapSPP. We show this by proving that $$O(\log n)$$O(logn)-approximate $$\textsf {GapSPP}$$GapSPP is in $$\mathsf {coNP} $$coNP.
In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an $$O(\sqrt{n})$$O(n) factor, improving upon the prior best factor of $$\omega (\sqrt{n \log n})$$ω(nlogn).

2016

CRYPTO

2015

EPRINT

2014

CRYPTO

2014

EPRINT

#### Program Committees

- TCC 2019
- Crypto 2018

#### Coauthors

- Divesh Aggarwal (1)
- Navid Alamati (1)
- Yevgeniy Dodis (4)
- Jianwei Li (1)
- Ilya Mironov (3)
- Tamalika Mukherjee (1)
- Phong Q. Nguyen (1)
- Chris Peikert (1)
- Adi Shamir (2)
- Daniel Wichs (2)