Decoding by Embedding: Correct Decoding Radius and DMT Optimality

The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP) are the core algorithmic problems on Euclidean lattices. They are central to the applications of lattices in many problems of communications and cryptography. Kannan's \emph{embedding technique} is a powerful technique for solving the approximate CVP, yet its remarkable practical performance is not well understood. In this paper, the embedding technique is analyzed from a \emph{bounded distance decoding} (BDD) viewpoint. We present two complementary analyses of the embedding technique: We establish a reduction from BDD to Hermite SVP (via unique SVP), which can be used along with any Hermite SVP solver (including, among others, the Lenstra, Lenstra and Lov\'asz (LLL) algorithm), and show that, in the special case of LLL, it performs at least as well as Babai's nearest plane algorithm (LLL-aided SIC). The former analysis helps to explain the folklore practical observation that unique SVP is easier than standard approximate SVP. It is proven that when the LLL algorithm is employed, the embedding technique can solve the CVP provided that the noise norm is smaller than a decoding radius $\lambda_1/(2\gamma)$, where $\lambda_1$ is the minimum distance of the lattice, and $\gamma \approx O(2^{n/4})$. This substantially improves the previously best known correct decoding bound $\gamma \approx {O}(2^{n})$. Focusing on the applications of BDD to decoding of multiple-input multiple-output (MIMO) systems, we also prove that BDD of the regularized lattice is optimal in terms of the diversity-multiplexing gain tradeoff (DMT), and propose practical variants of embedding decoding which require no knowledge of the minimum distance of the lattice and/or further improve the error performance.Comment: To appear in IEEE Transactions on Information Theor

Twisting Lattice and Graph Techniques to Compress Transactional Ledgers

International audienceKeeping track of financial transactions (e.g., in banks and blockchains) means keeping track of an ever-increasing list of exchanges between accounts. In fact, many of these transactions can be safely " forgotten " , in the sense that purging a set of them that compensate each other does not impact the network's semantic meaning (e.g., the accounts' balances). We call nilcatenation a collection of transactions having no effect on a network's semantics. Such exchanges may be archived and removed, yielding a smaller, but equivalent ledger. Motivated by the computational and analytic benefits obtained from more compact representations of numerical data, we formalize the problem of finding nilcatenations, and propose detection methods based on graph and lattice-reduction techniques. Atop interesting applications of this work (e.g., decoupling of centralized and distributed databases), we also discuss the original idea of a " community-serving proof of work " : finding nilcatenations constitutes a proof of useful work, as the periodic removal of nilcatenations reduces the transactional graph's size

A Gentle Tutorial for Lattice-Based Cryptanalysis

The applicability of lattice reduction to a wide variety of cryptographic situations makes it an important part of the cryptanalyst\u27s toolbox. Despite this, the construction of lattices and use of lattice reduction algorithms for cryptanalysis continue to be somewhat difficult to understand for beginners. This tutorial aims to be a gentle but detailed introduction to lattice-based cryptanalysis targeted towards the novice cryptanalyst with little to no background in lattices. We explain some popular attacks through a conceptual model that simplifies the various components of a lattice attack

An Improved BKW Algorithm for LWE with Applications to Cryptography and Lattices

In this paper, we study the Learning With Errors problem and its binary variant, where secrets and errors are binary or taken in a small interval. We introduce a new variant of the Blum, Kalai and Wasserman algorithm, relying on a quantization step that generalizes and fine-tunes modulus switching. In general this new technique yields a significant gain in the constant in front of the exponent in the overall complexity. We illustrate this by solving p within half a day a LWE instance with dimension n = 128, modulus $q = n^2$, Gaussian noise $\alpha = 1/(\sqrt{n/\pi} \log^2 n)$ and binary secret, using $2^{28}$ samples, while the previous best result based on BKW claims a time complexity of $2^{74}$ with $2^{60}$ samples for the same parameters. We then introduce variants of BDD, GapSVP and UniqueSVP, where the target point is required to lie in the fundamental parallelepiped, and show how the previous algorithm is able to solve these variants in subexponential time. Moreover, we also show how the previous algorithm can be used to solve the BinaryLWE problem with n samples in subexponential time $2^{(\ln 2/2+o(1))n/\log \log n}$. This analysis does not require any heuristic assumption, contrary to other algebraic approaches; instead, it uses a variant of an idea by Lyubashevsky to generate many samples from a small number of samples. This makes it possible to asymptotically and heuristically break the NTRU cryptosystem in subexponential time (without contradicting its security assumption). We are also able to solve subset sum problems in subexponential time for density $o(1)$, which is of independent interest: for such density, the previous best algorithm requires exponential time. As a direct application, we can solve in subexponential time the parameters of a cryptosystem based on this problem proposed at TCC 2010.Comment: CRYPTO 201

A Note on the Density of the Multiple Subset Sum Problems

It is well known that the general subset sum problem is NP-complete. However, almost all subset sum problems with density less than $0.9408\ldots$ can be solved in polynomial time with an oracle that can find the shortest vector in a special lattice. In this paper, we give a similar result for the multiple subset sum problems which has $k$ subset sum problems with the same solution. Some extended versions of the multiple subset sum problems are also considered. In addition, a modified lattice is involved to make the analysis much simpler than before