Master of Logic Project Report: Lattice Based Cryptography and Fully Homomorphic Encryption

Vectors and matrices are denoted by bold lowercase and uppercase letters. For purposes of matrix multiplication, all vectors are considered as column vectors. We denote the floor and ceiling functions by b·c and d·e respectively. Rounding to the closest integer (to the smaller one if there are two) is denoted by b·e. For q ∈ Z, let Zq = (bq/2c, bq/2c] ∩ Z and for x ∈ Z, let [x]q be the unique y ∈ Zq such that x = y+kq for some integer k. For vectors x ∈ Z, [x]q denotes the component-wise application of this operation. Note that here, Zq does not denote the ring Z/Zq. In particular, for x ∈ R and y ∈ Zq, x · y denotes the multiplication of x and y in R. The standard scalar product is denoted by 〈·, ·〉 and the euclidean norm by || · ||. Let ∼⊆ R × R be the relation defined by x ∼ y ⇔ ∃k ∈ Z : x + k = y. Let T = R/ ∼. For x ∈ R, we denote x/ ∼∈ T by x mod T. A function : N → R is called negligible if for every polynomial function p, there is some n0 ∈ N such that for all n ≥ n0, (n) < p(n)−1. A function δ : N → R is called overwhelming if there is a negligible function such that δ(n) ≥ 1− (n) for all n. For s > 0 and c ∈ R,let ρ s,c (x) = exp(π||(x− c)/s||). The total measure is ∫ x∈Rn ρs,c(x)dx = s n and the density function of the continuous Gaussian