## CryptoDB

### Thomas Espitau

#### Publications

**Year**

**Venue**

**Title**

2020

CRYPTO

Fast reduction of algebraic lattices over cyclotomic fields
📺
Abstract

We introduce a framework generalizing lattice reduction algorithms to module
lattices in order to \emph{practically} and \emph{efficiently} solve the
$\gamma$-Hermite Module-SVP problem over arbitrary cyclotomic fields. The core
idea is to exploit the structure of the subfields for designing a recursive
strategy of reduction in the tower of fields we are working in. Besides, we
demonstrate how to leverage the inherent symplectic geometry existing such
fields to provide a significant speed-up of the reduction for rank two
modules. As a byproduct, we also generalize to all cyclotomic fields and
provide speedups for many previous number theoretical algorithms, in
particular to the rounding in the so-called Log-unit lattice. Quantitatively,
we show that a module of rank 2 over a cyclotomic field of degree $n$ can be
heuristically reduced within approximation factor $2^{\tilde{O}(n)}$ in time
$\tilde{O}(n^2B)$, where $B$ is the bitlength of the entries. For $B$ large
enough, this complexity shrinks to $\tilde{O}(n^{\log_2 3}B)$. This last
result is particularly striking as it goes below the estimate of $n^2B$ swaps
given by the classical analysis of the LLL algorithm using the decrease of
the \emph{potential} of the basis. Finally, all this framework is fully parallelizable, and we
provide a full implementation. We apply it to break multilinear cryptographic
candidates on concrete proposed parameters. We were able to reduce matrices of
dimension 4096 with 6675-bit integers in 4 days, which is more than a million
times faster than previous state-of-the-art implementations. Eventually, we
demonstrate a quasicubic time for the Gentry-Szydlo algorithm which finds a
generator given the relative norm and a basis of an ideal. This algorithm is
important in cryptanalysis and requires efficient ideal multiplications and
lattice reductions; as such we can practically use it in dimension 1024.

2018

ASIACRYPT

LWE Without Modular Reduction and Improved Side-Channel Attacks Against BLISS
Abstract

This paper is devoted to analyzing the variant of Regev’s learning with errors (LWE) problem in which modular reduction is omitted: namely, the problem (ILWE) of recovering a vector $$\mathbf {s}\in \mathbb {Z}^n$$ given polynomially many samples of the form $$(\mathbf {a},\langle \mathbf {a},\mathbf {s}\rangle + e)\in \mathbb {Z}^{n+1}$$ where $$\mathbf { a}$$ and e follow fixed distributions. Unsurprisingly, this problem is much easier than LWE: under mild conditions on the distributions, we show that the problem can be solved efficiently as long as the variance of e is not superpolynomially larger than that of $$\mathbf { a}$$. We also provide almost tight bounds on the number of samples needed to recover $$\mathbf {s}$$.Our interest in studying this problem stems from the side-channel attack against the BLISS lattice-based signature scheme described by Espitau et al. at CCS 2017. The attack targets a quadratic function of the secret that leaks in the rejection sampling step of BLISS. The same part of the algorithm also suffers from a linear leakage, but the authors claimed that this leakage could not be exploited due to signature compression: the linear system arising from it turns out to be noisy, and hence key recovery amounts to solving a high-dimensional problem analogous to LWE, which seemed infeasible. However, this noisy linear algebra problem does not involve any modular reduction: it is essentially an instance of ILWE, and can therefore be solved efficiently using our techniques. This allows us to obtain an improved side-channel attack on BLISS, which applies to 100% of secret keys (as opposed to $${\approx }7\%$$ in the CCS paper), and is also considerably faster.

#### Coauthors

- Gilles Barthe (1)
- Sonia Belaïd (1)
- Jean-François Biasse (1)
- Jonathan Bootle (1)
- Claire Delaplace (1)
- Pierre-Alain Fouque (6)
- Alexandre Gélin (1)
- Benjamin Grégoire (1)
- Pierre Karpman (2)
- Paul Kirchner (2)
- Mélissa Rossi (1)
- Mehdi Tibouchi (2)