Incremental hash functions

Hash functions are one of the most important cryptographic primitives. They map an input of arbitrary finite length to a value of fixed length by compressing the input, that is why, they are called hash. They must run efficiently and satisfy some cryptographic security arguments. They are mostly used for data integrity and authentication such as digital signatures. Some hash functions such as SHA family (SHA1-SHA2) and MD family (MD2- MD4-MD5) are standardized to be used in cryptographic schemes. A common property about their construction is that they are all iterative. This property may cause an efficiency problem on big size data, because they have to run on the entire input even it is slightly changed. So the question is "Is it possible to reduce the computational costs of hash functions when small modifications are done on data?" In 1995, Bellare, Goldreich and Goldwasser proposed a new concept called incrementality: a function f is said to be incremental if f(x) can be updated in time proportional to the amount of modification on the input x. It brings out two main advantages on efficiency: incrementality and parallelizability. Moreover, it gives a provable security depending on hard problems such as discrete logarithm problem (DLP). The hash functions using incrementality are called Incremental Hash Functions. Moreover, in 2008, Dan Brown proposed an incremental hash function called ECOH by using elliptic curves, where DLP is especially harder on elliptic curves, and which are therefore quite popular mathematical objects in cryptography. We state incremental hash functions with some examples, especially ECOH , and give their security proofs depending on hard problems.