Good question, but scientist have not been able to even separate "Algorithmica" from the remaining cases, let alone decide the exact world we live in.
That said, there are several research papers on the subject. See for example:
On the possibility of basing Cryptography on the assumption that P != NP by Goldreich and Goldwasser, and references thereof.
See also On basing one-way functions on NP-hardness by Adi Akavia et al.
In addition, it is well known that decoding some cryptosystems is NP-hard (See, for example, the McEliece cryptosystem, or Lattice-based cryptography).
I don't know why this does NOT resolve the problem, since I'm not familiar with such cryptosystems.See Peter Shor comments below.
I also suggest you take a quick look at the discussion at Stackoverflow. Reviewing the literature which cites Impagliazzo's work can also be instructive.
The following results might be of interest:
Feigenbaum and Fortnow. Random-Self-Reducibility of Complete Sets.
SIAM Journal on Computing, 22:994–1005, 1993.
Bogdanov and Trevisan. On Worst-Case to Average-Case Reductions for NP
Problems. In Proceedings of the 44th Annual IEEE Symposium on Foundations
of Computer Science, pages 308–317, 2003.
Akavia, Goldreich, Goldwasser, and Moshkovitz. On Basing One-Way Functions on NP-Hardness
Gutfreund and Ta-Shma. New connections between derandomization, worst-case complexity and average-case complexity. Tech. Rep. TR06-108, Electronic Colloquium on Computational Complexity, 2006.
Bogdanov and Trevisan. Average-case complexity. Found. Trends Theor. Comput. Sci. 2, 1 (Oct. 2006), 1-106. DOI= http://dx.doi.org/10.1561/0400000004