What does the linear assumption over bilinear groups mean? In the abstract of "Cryptography with Tamperable and Leaky Memory", at the end of the 3rd paragraph, the authors say: 


  In both schemes we rely on the linear assumption over bilinear groups.


What is the linear assumption ? How do you break this linear assumption ? 

I would appreciate if somebody could give me concrete examples. More specifics about the paper where they talk about linear assumption:
The problem you are referring to seems to be the Decisional Linear Assumption (DLIN), which states that given $(u,v,u^a,v^b)\in \mathbb{G}^4$, it is hard to distinguish a couple $(h,h^{a+b}) \in \mathbb{G}^2$ from a totally random couple $(h,h') \in \mathbb{G}^2$.

There is also the Computational Linear Assumption (CLIN), which states that it is hard to  compute $h^{a+b}$ given $(u,v,u^a,v^b,h)$. 

The Decisional Linear Assumption is a weaker assumption (in the sense that it's harder to break) than Decisional Diffie-Hellman Assumption (DDH), so it can come in handy when DDH does not hold, which often happens in pairing-based cryptography.
According to this paper (page 9), we even have the following reduction :
$$DDH &lt; DLIN &lt; CLIN &lt; CDH$$

I tried to sum up what I found but I'm not an expert in pairing-based crypto so I hope everything was correct.