Non-adaptive Universal One-Way Hash Functions from Arbitrary One-Way Functions

In this work we give the first non-adaptive construction of universal one-way hash functions (UOWHFs) from arbitrary one-way functions. Our construction uses $O(n^9)$ calls to the one-way function, has a key of length $O(n^{10})$, and can be implemented in NC1 assuming the underlying one-way function is in NC1.

Prior to this work, the best UOWHF construction used O(n13) adaptive calls and a key of size O(n5) (Haitner, Holenstein, Reingold, Vadhan and Wee [Eurocrypt ’10]). By the result of Applebaum, Ishai and Kushilevitz [FOCS ’04], the above implies the existence of UOWHFs in NC0, given the existence of one-way functions in NC1.

We also show that the PRG construction of Haitner, Reingold and Vadhan (HRV, [STOC ’10]), with small modifications, yields a relaxed notion of UOWHFs , which is a function family which can be (inefficiently) converted to UOWHF by changing the functions on a negligible fraction of the inputs. In order to analyze this construction, we introduce the notion of next-bit unreachable entropy, which replaces the next-bit pseudoentropy notion used by HRV.