We study the following two kinds of one-way hash functions: universal oneway hash functions (UOHs) and collision intractable hash functions (CIHs). The main property of the former is that given an initial-string x, it is computationally difficult to find a different string y that collides with x. And the main property of the latter is that it is computationally difficult to find a pair x � = y of strings such that x collides with y. Our main results are as follows. First we prove that UOHs with respect to initial-strings chosen arbitrarily exist if and only if UOHs with respect to initial-strings chosen uniformly at random exist. Then, as an application of the result, we show that UOHs with respect to initial-strings chosen arbitrarily can be constructed under a weaker assumption, the existence of one-way quasi-injections. Finally, we investigate relationships among various versions of one-way hash functions. We prove that some versions of one-way hash functions are strictly included in others by explicitly constructing hash functions that are one-way in the sense of the former but not in the sense of the latter. 1

This paper reveals a duality between constructions of two basic cryptographic primitives, pseudo-random string generators and one-way hash functions. Applying the duality, we present a construction for universal one-way hash functions assuming the existence of one-way permutations. Under a stronger assumption, the existence of distinction-intractable permutations, we prove that the construction constitutes a collision-intractable hash function. Using ideas behind the construction, we propose practical one-way hash functions, the fastest of which compress nearly 2n-bit long input into n-bit long output strings by applying only twice a one-way function. 1