To gain strong confidence in the security of a public-key scheme, it is most desirable for the security proof to feature a tight reduction between the adversary and the algorithm solving the under-lying hard problem. Recently, Chen and Wee (Crypto '13) described the first Identity-Based Encryption scheme with almost tight security under a standard assumption. Here, "almost tight" means that the security reduction only loses a factor O(λ) – where λ is the security parameter – instead of a factor proportional to the number of adversarial queries. Chen and Wee also gave the shortest signatures whose security almost tightly relates to a simple assumption in the standard model. Also recently, Hofheinz and Jager (Crypto '12) constructed the first CCA-secure public-key encryption scheme in the multi-user setting with tight security. These constructions give schemes that are significantly less efficient in length (and thus, processing) when compared with the earlier schemes with loose reductions in their proof of security. Hofheinz and Jager's scheme has a ciphertext of a few hundreds of group elements, and they left open the problem of finding truly efficient constructions. Likewise, Chen and Wee's signatures and IBE schemes are somewhat less efficient than previous constructions with loose reductions from the same assumptions. In this paper, we consider space-efficient schemes with security almost tightly related to standard assumptions. As a step in solving the open question by Hofheinz and Jager, we construct an ef-ficient CCA-secure public-key encryption scheme whose chosen-ciphertext security in the multi-challenge, multi-user setting almost tightly relates to the DLIN assumption (in the standard model). Quite remark-ably, the ciphertext size decreases to 69 group elements under the DLIN assumption whereas the best previous solution required about 400 group elements. Our scheme is obtained by taking advantage of a new almost tightly secure signature scheme (in the standard model) we develop here and which is based on the recent concise proofs of linear subspace membership in the quasi-adaptive non-interactive zero-knowledge setting (QA-NIZK) defined by Jutla and Roy (Asiacrypt '13). Our signature scheme reduces the length of the previous such signatures (by Chen and Wee) by 37% under the Decision Linear assump-tion, by almost 50% under the K-LIN assumption, and it becomes only 3 group elements long under the Symmetric eXternal Diffie-Hellman assumption. Our signatures are obtained by carefully combining the proof technique of Chen and Wee and the above mentioned QA-NIZK proofs.