GTACS

Fully homomorphic encryption (FHE) provides a simple template for secure computation between two parties (Alice and Bob) where: (I) Alice encrypts her input under her key, (II) Bob homomorphically evaluates the desired function on Alice's ciphertext and his own input, and sends the encrypted output to Alice.

A protocol for computing a functionality is secure if an adversary in this protocol cannot cause more harm than in an ideal computation,
where parties give their inputs to a trusted party which returns the output of the functionality to all parties.
In particular, in the ideal model such computation is fair -- if the corrupt parties get the output, then the honest parties get the output.
Cleve (STOC 1986) proved that, in general, fairness is not possible without an honest majority.

Secure Multiparty Computation for the case of dishonest majority
has previously been known as the case where no efficient solution
was possible, since here one cannot avoid using expensive public-key
machinery. However, in a recent of line of research it has been shown
that all the hard work can be pushed into a preprocessing phase
that is independent of the function to be computed. Then, in an
on-line phase, one can compute the function very efficiently
using only cheap information theoretic primitives.