Fully homomorphic encryption (FHE) techniques are capable of
performing encrypted computation on Boolean circuits, i.e., the
user specifies encrypted inputs to the program, and the server computes on the encrypted inputs. Applying these techniques to general programs with recursive procedures and data-dependent loops
has not been a focus of attention. In this paper, we take a first
step toward building a compiler that, given programs with complex
control flow, generates efficient code suitable for the application of
We first describe how programs written in a small Turing-complete instruction set can be executed with encrypted data and
point out inefficiencies in this methodology. We then provide examples of transforming (a) the greatest common divisor (GCD)
problem using Euclid\'s algorithm and (b) the 3-Satisfiability
(3SAT) problem using a recursive backtracking algorithm into a
path-levelized form to which FHE can be applied. We describe
how path levelization reduces control flow ambiguity and improves
encrypted computation efficiency. Using these techniques and data-dependent loops as a starting point, we then build support for hierarchical programs made up of phases, where each phase corresponds to a fixed point computation that can be used to further improve the efficiency of encrypted computation.
In our setting, the adversary learns an estimate of the number of
steps required to complete the computation, which we show is the
least amount of leakage possible.