[littlestymaar]

Here's my attempt:

Lattice-based crypto:

Regular public-key crypto mostly use one kind of mathematical structures (finite fields) who have a known hard problem (finding the discrete logarithm of a number of a finite field) while the reverse problem (calculate the power of n of a given number) is trivial. You can do crypto on every structure for which it exists such a problem.

Latice are different mathematical structures, for which it exists different kinds of problems that can be used for crypto.

secure multiparty computation

Let say you have some confidential data X and I have some confidential data Y. What we want to do is compute f(X,Y) without sharing the X and Y because we don't trust one another (or any third party).

I hope it helped.