One-way Functions, Commitments, and Collision Resistant Hashing

One-way functions, functions that are easy to compute, but hard to invert on average, are the most basic cryptographic application of lattices. Lattice-based one-way functions have some useful homomorphic properties that immediately yield useful cryptographic applications, like collision resistant hash functions, and commitment schemes.

Collision-Free Hashing from Lattice Problems
(Goldreich, Goldwasser & Halevi - Studies in Complexity and Cryptography 2011 / ECCC 1996)

More Efficient Commitments from Structured Lattice Assumptions
(Baum, Damgard, Lyubashevsky, Oechsner & Peikert - SCN 2018)