Notes mostly taken from [1].
One of the most important cryptographic assumptions is DDH. It says: ([a], [r], [ar]) is computationally indistinguishable from ([a], [r], [z]) where a, r, z are uniform elements from Z_q and [a] means a*g (written additively) or g^a (written multiplicatively) and g is an element of a cyclic group of prime order q.
In bilinear groups DDH is no longer true - [ar] is no longer pseudorandom since e([a], [r]) = e(1, [ar]).
Boneh, Boyen, and Shacham [2] defined an alternative decisional assumption for G1 - given u, v, h, z from G1, (u, v, h, u^a, v^b, h^(a+b)) is computationally indistinguishable from (u, v, h, u^a, v^b, h^z). This is called (two-) linear decision assumption.
It can be written also: ([a1], [a2], [a1r1], [a2r2], [r1+r2]) is computationally indistinguishable from ([a1], [a2], [a1r1], [a2r2], [z]). It can be generalized to k-lin assumption.
It is useful to see DDH assumption as a hard subgroup membership problem in G^2. DDH then says that it is difficult to decide whether ([b], [c]) is in a group generated by ([1], [a]).
Strong Diffie-Hellman assumption [3] is similar to the Strong RSA assumption.
Recall that Strong RSA assumption says: given an RSA modulus N and s from Z_n*, it is difficult to find (a, e) such that a^e = s (mod N).
Strong Diffie-Hellman assumption (q-SDH) says: given g, g^x,..., g^(x^q), it is difficult to find (a, c) such that a^(x+c) = g.
[1] Escala, Alex, et al. "An algebraic framework for Diffie–Hellman assumptions." Journal of cryptology 30.1 (2017): 242-288.
[2] Boneh, Dan, Xavier Boyen, and Hovav Shacham. "Short group signatures." Annual International Cryptology Conference. Springer, Berlin, Heidelberg, 2004.
[3] Boneh, Dan, and Xavier Boyen. "Short signatures without random oracles." International Conference on the Theory and Applications of Cryptographic Techniques. Springer, Berlin, Heidelberg, 2004.