We prove that the module learning with errors (M-LWE) problem with arbitrary polynomial-sized modulus p is classically at least as hard as standard worst-case lattice problems, as long as the module rank d is not smaller than the number field degree n. Previous publications only showed the hardness under quantum reductions. We achieve this result in an analogous manner as in the case of the learning with errors (LWE) problem. First, we show the classical hardness of M-LWE with an exponential-sized modulus. In a second step, we prove the hardness of M-LWE using a binary secret. And finally, we provide a modulus reduction technique. The complete result applies to the class of powerof-two cyclotomic fields. However, several tools hold for more general classes of number fields and may be of independent interest.

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We prove that the Module Learning With Errors (M-LWE) problem with binary secrets and rank d is at least as hard as the standard version of M-LWE with uniform secret and rank k, where the rank increases from k to d ≥ (k + 1) log 2 q + ω(log 2 n), and the Gaussian noisewhere n is the ring degree and q the modulus. Our work improves on the recent work by Boudgoust et al. in 2020 by a factor of √ md in the Gaussian noise, where m is the number of given M-LWE samples, when q fulfills some number-theoretic requirements. We use a different approach than Boudgoust et al. to achieve this hardness result by adapting the previous work from Brakerski et al. in 2013 for the Learning With Errors problem to the module setting. The proof applies to cyclotomic fields, but most results hold for a larger class of number fields, and may be of independent interest.

The Fiat-Shamir with Aborts paradigm of Lyubashevsky has given rise to efficient lattice-based signature schemes. One popular implementation is Dilithium, which has been selected for standardization by the US National Institute of Standards and Technology (NIST). Informally, it can be seen as a lattice analog of the well-known discrete-logarithm-based Schnorr signature. An interesting research question is whether it is possible to combine several unrelated signatures, issued from different signing parties on different messages, into one single aggregated signature. Of course, its size should be significantly smaller than the trivial concatenation of all signatures. Ideally, the aggregation can be done offline by a third party, called public aggregation. Previous works have shown that it is possible to half-aggregate Schnorr signatures, but it was left open if the underlying techniques can be adapted to the lattice setting. In this work, we show that, indeed, we can use similar strategies to obtain a signature scheme allowing for public aggregation whose hardness is proven assuming the intractability of well-studied problems on module lattices. Unfortunately, our scheme produces aggregated signatures that are larger than the trivial solution of concatenating. This is due to peculiarities that seem inherent to lattice-based cryptography. Its motivation is thus mainly pedagogical.

At CRYPTO 2017, Roşca et al. introduce a new variant of the Learning With Errors (LWE) problem, called the Middle-Product LWE (MP-LWE). The hardness of this new assumption is based on the hardness of the Polynomial LWE (P-LWE) problem parameterized by a set of polynomials, making it more secure against the possible weakness of a single defining polynomial. As a cryptographic application, they also provide an encryption scheme based on the MP-LWE problem. In this paper, we propose a deterministic variant of their encryption scheme, which does not need Gaussian sampling and is thus simpler than the original one. Still, it has the same quasi-optimal asymptotic key and ciphertext sizes. The main ingredient for this purpose is the Learning With Rounding (LWR) problem which has already been used to derandomize LWE type encryption. The hardness of our scheme is based on a new assumption called Middle-Product Computational Learning With Rounding, an adaption of the computational LWR problem over rings, introduced by Chen et al. at ASIACRYPT 2018. We prove that this new assumption is as hard as the decisional version of MP-LWE and thus benefits from worst-case to average-case hardness guarantees.