Homomorphic encryption has progressed rapidly in both efficiency and versatility since its emergence in 2009. Meanwhile, a multitude of pressing privacy needs --- ranging from cloud computing to healthcare management to the handling of shared databases such as those containing genomics data --- call for immediate solutions that apply fully homomorpic encryption (FHE) and somewhat homomorphic encryption (SHE) technologies. Recent rapid progress in fully homomorphic encryption has catalyzed renewed efforts to develop efficient privacy preserving protocols. Several works have already appeared in the literature that provide solutions to these problems by employing leveled or somewhat homomorphic encryption techniques. Here, we propose efficient ways of adapting the most fundamental programming problems; boolean algebra, arithmetic in binary and higher radix representation, sorting, and search to the fully homomorphic encryption domain by focusing on the multiplicative depth of the circuits alongside the more traditional metrics. The reduced depth allows much reduced noise growth and thereby makes it possible to select smaller parameter sizes in leveled FHE instantiations resulting in greater efficiency savings. We begin by exploring already existing solutions to these programming problems, and analyze them in terms of homomorphic evaluation and memory costs. Most of these algorithms appear to be not the best candidates for FHE solutions, hence we propose new methods and improvements over the existing algorithms to optimize performance.