Suppose we want to compare two protocols on the computational costs. One of the protocols is designed using random oracle model (we call it the first protocol) and the other protocol uses is designed over common reference string (we call it the second protocol).
Now if we want to compare these two, we have to rely on the most costly operation in them. In the first protocol it is the hashing operation and in the second, the modular exponentiation. How can I compare these two protocols assuming both of them work over the same length of security parameter.
For simplicity, assume that the first protocol uses "n" hashing operation to complete and the second protocol uses "2n" modular exponentiation to output the result.
Is there any way to compare these two? What measures should I consider when comparing these protocols?
To give a little background on the models, the Common Reference String (CRS) is the model which a global coin is assumed, which means a reference string (mostly a random one) is distributed among the parties. The random oracle model is the model which is essentially formalizing the hash functions. In this model, every party has oracle access to a random function (i.e., they submit the query to a function (oracle) some where and get the results, which in this case is a random string). When implementing a protocol with this model, the queries are replaced by calculating a hash function.