Optimizations in Fully Homomorphic Encryption

Optimizations in fully homomorphic encryption are the guidelines of many cryptographic researches after Gentry's breakthrough in 2009. In this chapter, we sketch different technics to optimize and simplify fully homomorphic encryption schemes. Among these technics, we will focus on the trans-ciphering one, for this method we will describe a homomorphic evaluation of the different AES circuits (AES-128, AES-192 and AES-256) using a noise-free fully homomorphic encryption scheme. After all, we will present a new noise-free fully homomorphic encryption scheme based on quaternions. Trans-ciphering is supposed to be an efficient solution to optimize data storage in a context of outsourcing computations to a remote cloud computing as it is considered a powerful tool to minimize runtime in the client side. In this implementation, we will use our noise-free fully homomorphic encryption scheme with different key sizes. Among the tools we are using in this work, a small laptop with characteristics: bi-cores Intel core i5 CPU running at 2.40 GHz, with 512 KB L2 cache and 4 GB of Random Access Memory. Our implementation takes about 18 min to evaluate an entire AES circuit using a key of 1024 bits for the fully homomorphic encryption scheme.