Machine learning on encrypted data ...The efficiency of different encoding choices is measured with three metrics: The number of additions we need to perform in the underlying plaintext space for a given computation on the rational numbers the number of multiplications in the plaintext space and the multiplicative depth. The latter measures the number of consecutive multiplications in the plaintext space needed to perform a computation on rational numbers and is motivated by the concrete structure of the Fully Homomorphic Encryption schemes we have today. In this work we first show in Chapter 2 that among all finite fields GF(p^k) when adding or multiplying two natural numbers the choice GF(2) is best in terms of the number of additions and multiplications. In terms of multiplicative depth there is no generic optimum as this depends on the concrete function and the input length of the function arguments. However we do show that choosing k>1 always has worse performance than choosing GF(p). Because of this finding we focus on the encoding base GF(2) in the rest of the work. In Chapter 3 we extend our analysis to include negative numbers and thus examine the effort incurred by the two most popular encoding for signed numbers Twos Complement and Sign Magnitude. We see that Twos Complementis better for adding and Sign Magnitude is better for multiplying two numbers. We utilize this fact to invent a new encoding called Hybrid Encoding which essentially switches between the two to minimize the effort. Our new encoding induces a performance gain of over 70% in some of our applications. We then extend our analysis from integer to rational numbers in Chapter 4. We propose several optimizations which result in an efficiency gain of over 95%. We also propose ways to speed up the comparison function...