As per the recently released The Stability and the Security of the Tangle, a 2018 ICUBE - University of Strasbourg "study of the stability and the security of the distributed data structure at the base of the IOTA protocol, called the Tangle", a proof is provided that either:
(1) All the honest nodes must constantly use all their hashing power to validate the main chain (similarly to the bitcoin protocol)
(2) Or some kind of authority must be provided to avoid this kind of attack (like in the current version of the IOTA where a coordinator is used).
Skipping to the conclusion of this landmark study on the tangle reveals the following conclusion:
We presented a model to analyze the Tangle and we used it to study the average confirmation time and the average number of unconfirmed transaction over the time.
Then, we defined the notion of assiduous honest majority that captures the fact that the honest nodes have more hashing power than the adversarial nodes and that all this hashing power is constantly used to create transactions. We proved that for any tip selection algorithm that has a maximal deterministic tip selection (which is the case for all currently known TSA), the assiduous honest majority assumption is necessary to prevent a double-spending attack on the Tangle.
Our analyze shows that honest nodes cannot stay at rest, and should be continuously signing transactions (even empty ones) to increase the weight of their local main sub-DAG. If not, their available hashing power cannot be used to measure the security of the protocol, like we see for the Bitcoin protocol. Indeed, having a huge number of honest nodes with a very large amount of hashing power cannot prevent an adversary from attacking the Tangle if the honest nodes are not using this hashing power. This conclusion may seem intuitive, but the fact that it is true for all tip selection algorithms (that have a deterministic maximal TSA) is something new that have not been proved before.
I would like to highlight the following profound observations:
for any tip selection algorithm ... the assiduous honest majority assumption is necessary to prevent a double-spending attack on the Tangle.
Our analyze shows that honest nodes cannot stay at rest, and should be continuously signing transactions (even empty ones) to increase the weight of their local main sub-DAG.
If not, their available hashing power cannot be used to measure the security of the protocol, like we see for the Bitcoin protocol
having a huge number of honest nodes with a very large amount of hashing power cannot prevent an adversary from attacking the Tangle if the honest nodes are not using this hashing power
Given that IOTA is committed to removing The Coordinator, how will IOTA ensure that all honest nodes are continuously using their hashing power?
@ben75 has answered that the 'Assiduous Honest Majority' must not be misunderstood. Here it is for clarification and for the record:
Assiduous Honest Majority Assumption
The cumulative weight and the score can be used by a node to select its main DAG. However, even if it is true that a heavy sub-DAG is harder to generate than a light one, there is no relation yet in the protocol between the weight of sites and the hashing power capacity of honest nodes.
We define the assiduous honest majority assumption as the fact that the hashing power of honest nodes is constantly used to generate sites and that it is strictly greater than the hashing power of the adversary. In fact, without this assumption, it is not relevant to look at the hashing power of the honest nodes if they do not constantly use it to generates new sites.
Thus, under this assumption, the cumulative weight of the honest DAG grows according to the hashing power of the honest nodes, and the probability that an adversary generates more sites than the honest nodes in a given period of time tends to 0 as the duration of the period tends to infinity. Conversely, without this assumption, an adversary may be able to generates more sites than the honest nodes, even with less available hashing power
The document is also available through this source: https://hal.archives-ouvertes.fr/hal-01716111v2