Reading about Accountability in ABE currently and the paper "White-box traceable ciphertext-policy attribute-based encryption supporting flexible attributes" when comparing its system with an existing mentioned the fact that theirs is a prime order group scheme while the other is a composite order group scheme. My question is what significance does this have on the underlying scheme. And are there cases where one is preferred to the other or cases where on is applicable and the other is not.
Composite-order groups for pairings are sometimes a necessary tool provide a proof for adaptive security. Most schemes for prime-order groups only reach selective security in their respective security models.
While the security generally increases, the performance of composite-order group pairings decreases heavily. Compare prime-order pairings of Type A with composite-order pairings of Type A1 here. All operations for Type A1 take roughly 18 times as much time as operations for Type A (although, they have different Dlog security levels, but that doesn't influence as much).
Dual Pairing Vector Spaces (over prime-order groups) behave similar to composite-order pairings in that they enable more powerful security proofs, but perform roughly 10 times slower than basic prime-order pairings.