Lattice cryptography offers the (over)generalized take-home message that indeed one can design cryptosystems where breaking the average case is as hard as solving a particular NP-hard problem (typically the Shortest Vector Problem or the Closest Vector Problem).
I can recommend reading the introduction section of http://eprint.iacr.org/2008/521 and then chasing references to the cryptosystems.
Also, see the lecture notes at http://www.cs.ucsd.edu/~daniele/CSE207C/, and chase links for a book if you want.
