Quadratic scaling of hashing time
In Bitcoin a transaction moves money from one address to another and in order to only allow the actual owner of money the ability to move this, we use cryptographic signatures.
An owner of a Bitcoin address has to add a cryptographic signature inside of that transaction that spends his money.
There is a design flaw in Transactions v1, as Satoshi created it, which can lead to the validation of signatures taking much longer than expected.
The issue is that a transaction with more than one input defines that the thing that is signed is not the same for each of those inputs.
The cryptographic concept of signing is a two step process, one is to read and process the entire transaction with the output of first step being signed as a second step. This means that a transaction with 10 inputs will have a minimum of 10 check-signature commands. Each of those will hash the entire transaction, with minor differences.
A natural consequence of this is that adding another input will not only add one processing of the entire transaction, it will also make each already existing check-signature command process a larger transaction. Making the amount of work grow quadratically.
Isn't checking signatures incredibly fast?
We have a pretty good implementation for hashing in Bitcoin Classic, although there is certainly room for improvement, and we can hash at hundreds of megabytes every second.
The problem written here is one that makes the amount of data we have to actually process grow quadratically based on the size of the transaction and the number of inputs. Some attacks are making it so that a 1MB block, containing 2 transactions, can end up hashing that one MB 19,100 times.
This is probably the worst case scenario in a 1MB transaction limit, but it is a very good demonstration of the problem. Notice that bigger blocks will not cause there to be bigger transactions which in turn means that bigger blocks will not cause quadratic slowdowns.
How does Flexible Transactions solve this?
The solution is rather simple and elegant, we replaced repeated hashing of the entire transaction with using the transaction-ID (and some other parts) as the input of a signature. This means that the size of the transaction no longer is relevant and it only needs to be calculated once, regardless of the amount of inputs.
This is the same solution that solves the malleability issue.