To gain strong confidence in the security of a public-key scheme, itis most desirable for the security proof to feature a \\emph{tight}

reduction between the adversary and the algorithm solving the

underlying 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 $O(\\lambda)$

-- where $\\lambda$ 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 efficient

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

remarkably, 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 assumption, 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.