  Fully Homomorphic Encryption (FHE) is one of the most promising technologies
for privacy protection as it allows an arbitrary number of function
computations over encrypted data. However, the computational cost of these FHE
systems limits their widespread applications. In this paper, our objective is
to improve the performance of FHE schemes by designing efficient parallel
frameworks. In particular, we choose Torus Fully Homomorphic Encryption (TFHE)
as it offers exact results for an infinite number of boolean gate (e.g., AND,
XOR) evaluations. We first extend the gate operations to algebraic circuits
such as addition, multiplication, and their vector and matrix equivalents.
Secondly, we consider the multi-core CPUs to improve the efficiency of both the
gate and the arithmetic operations. Finally, we port the TFHE to the Graphics
Processing Units (GPU) and device novel optimizations for boolean and
arithmetic circuits employing the multitude of cores. We also experimentally
analyze both the CPU and GPU parallel frameworks for different numeric
representations (16 to 32-bit). Our GPU implementation outperforms the existing
technique, and it achieves a speedup of 20x for any 32-bit boolean operation
and 14.5x for multiplications.
