The Module Learning With Errors ( $$\text {M-LWE}$$ ) problem is a core computational assumption of lattice-based cryptography which offers an interesting trade-off between guaranteed security and concrete efficiency. The problem is parameterized by a secret distribution as well as an error distribution. There is a gap between the choices of those distributions for theoretical hardness results (standard formulation of $$\text {M-LWE}$$ , i.e., uniform secret modulo q and Gaussian error) and practical schemes (small bounded secret and error). In this work, we make progress toward narrowing this gap. More precisely, we prove that $$\text {M-LWE}$$ with uniform $$\eta $$ -bounded secret for any $$1 \le \eta \ll q$$ and Gaussian error, in both its search and decision variants, is at least as hard as the standard formulation of $$\text {M-LWE}$$ , provided that the module rank d is at least logarithmic in the ring degree n. We also prove that the search version of $$\text {M-LWE}$$ with large uniform secret and uniform $$\eta $$ -bounded error is at least as hard as the standard $$\text {M-LWE}$$ problem, if the number of samples m is close to the module rank d and with further restrictions on $$\eta $$ . The latter result can be extended to provide the hardness of search $$\text {M-LWE}$$ with uniform $$\eta $$ -bounded secret and error under specific parameter conditions. Overall, the results apply to all cyclotomic fields, but most of the intermediate results are proven in more general number fields.

We present a probabilistic polynomial-time reduction from the lattice Bounded Distance Decoding (BDD) problem with parameter 1/( sqrt(2) * gamma) to the unique Shortest Vector Problem (uSVP) with parameter gamma for any gamma > 1 that is polynomial in the lattice dimension n. It improves the BDD to uSVP reductions of [Lyubashevsky and Micciancio, CRYPTO, 2009] and [Liu, Wang, Xu and Zheng, Inf. Process. Lett., 2014], which rely on Kannan's embedding technique. The main ingredient to the improvement is the use of Khot's lattice sparsification [Khot, FOCS, 2003] before resorting to Kannan's embedding, in order to boost the uSVP parameter.

The most widely accepted models in the security proofs of authenticated key exchange protocols are the Canetti–Krawczyk (CK) and extended CK models that admit different adversarial queries with ambiguities and incomparable strength. It is desirable to incorporate specific and powerful adversarial queries into a single unified security model and establish a more practical oriented security notion. Concerning the security of one‐round implicitly authenticated Diffie–Hellman (DH) key exchange protocols, the authors present a unified security model that has many advantages over the previous ones. In the model, a system environment is set up, all of adversarial queries are practically interpreted and definitely characterised through physical environment, and some rigorous rules of secret leakage are also specified. To demonstrate usability of their model, a new protocol based on the OAKE protocol is proposed, which satisfies the presented strong security notion and attains high efficiency. The protocol is proven secure in random oracle model under gap DH assumption.