Abstract Verifiable encryption enables the decryption to be taken on properly generated ciphertexts, by making the encryptor provide a zero-knowledge proof. To meet the quantum-safe application requirements, such as key escrow, Lyubashevsky et al. proposed a one-shot verifiable encryption (LN17 scheme) based on the hardness of lattice problems. In their scheme, the FSwA-type zero-knowledge proof was obtained using rejection sampling on a discrete Gaussian distribution. In this paper, we present a construction of verifiable encryption that utilizes rejection sampling on bimodal Gaussian to get the associated zero-knowledge proof. Our new construction, while exhibiting a weaker soundness property than LN17 scheme, benefits from a smaller proof size, leading to a reduced size of the verifiable ciphertext. As for the weaker soundness property, it supports some applications such as key escrow where honestly generated verifiable ciphertexts are more useful to be decrypted out in the hope of doing some further computation tasks. We provide the efficiency comparison of the new construction by instantiating it with several sets of concrete parameters.

The cooperative system of multiple Autonomous Underwater Vehicles (AUVs) is becoming increasingly popular in environment survey, target search and many other marine coverage tasks. Coverage path planning is essential prior to task implementation to avoid path redundancy or area omission. A significant issue is that most existing underwater coverage path planning studies assume that AUVs can always obtain accurate position estimates, without considering the problem of positioning error divergence in underwater navigation systems. This article proposes a coverage path planning method based on Deep Reinforcement Learning (DRL) for the leader-follower AUVs cooperative navigation mode. The method ensures that the AUV formation successfully completes the coverage task while utilizing distance measurements between the leader and follower AUVs to achieve cooperative position estimation with bounded errors. Grid division strategy is used to ensure a close distance between the leader and following AUVs. The Double Deep QNetwork (DDQN) learning algorithm is applied for global cooperative coverage path planning, with a prior positioning uncertainty dictionary built based on the relationship between positioning errors and cooperative paths. An online path replanning method is also designed to avoid unknown static and dynamic obstacles. The proposed method is validated through simulations and lake experiments using unmanned surface vehicles. Compared to existing full-coverage path planning strategies, it demonstrates significant improvements in positioning accuracy, enabling AUVs to maintain high-precision navigation and accurately track the planned path during task execution.