Efficient FHE from (Standard) LWE

As anounced in the title, Brakerski and Vaikuntanathan (TODO: cite) introduced a fully homomorphic encryption scheme which is based only on the LWE assumption. They show that the security of the scheme can be reduced to the worst-case hardness of short vector problems on arbitrary latices.

They start by introducing a somewhat homomorphic encryption scheme SH, which is then transformed into a bootstrappable scheme BTS. In doing so, the authors deviate from previously known techniques of “squashing” the decryption algorithm of SH that has been used by Dijk, Gentry, Halevi and Vaikuntanathan in FHE over the integers and instead introduce a new “dimension-modulus reduction” technique, which shortens the ciphertexts and simplifies the decryption circuit of SH. This is all achieved without introducing additional assumptions, such as the hardness of the sparse subset-sum problem.

This scheme has particularly short ciphertexts and for this reasons the authors managed to use it for constructing an efficient single-server private information retrieval (PIR) protocol.

A Somewhat Homomorphic encryption Scheme SH

Let us start by presenting a somewhat homomorphic encryption scheme. Denote by $\lambda$ the security parameter. The scheme is parameterized by a dimension $n\in \mathbb {N}$, a positive integer $m\in \mathbb {N}$ , an odd modulus $q$ (which does not have to be prime) and a noise distribution $\chi$ over $\mathbb {Z} _{q}$. An additional parameter of the scheme $L\in \mathbb {N}$ is an upper bound on the maximal multiplicative depth that the scheme can homomorphically evaluate.

We are not going to elaborate on the appropriate choice of the size for the parameters, but we invite the curious reader to look at the paper. However, for brevity, we mention that the dimension $n$ is polynomial in the security parameter $\lambda$, $m\geq n\log(q)+2\lambda$ is a polynomial in $n$, the modulus is an odd number $q\in [2^{n^{\epsilon }},2\cdot 2^{n^{\epsilon }})$ is sub-exponential in $n$, i.e. $\epsilon$ is a positive constant which is strictly smaller than $1$. The noise distribution $\chi$ produces small samples, of magnitude at most $n$ in $\mathbb {Z} _{\mathfrak {q}}$. The depth bound $L$ is approximately of the size $\epsilon \cdot \log(n)$.

SH.Keygen: sample $L+1$ vectors $s_{0},\dots ,s_{L}$ uniformly from $\mathbb {Z} _{q}^{n}$ and compute, for all $l\in [L]$, $0\leq I\leq j\leq n$ and $\tau \in \{0,\dots ,\lfloor \log {q}\rfloor \}$, the value

$\psi _{l,i,j,\tau }:=\left(a_{l,i,j,\tau },b_{l,i,j,\tau }:=\langle a_{l,i,j,\tau },s_{l}\rangle +2\cdot e_{l,I,j,\tau }+2^{\tau }\cdot s_{l-1}\cdot s_{l-1}[j]\right)\in \mathbb {Z} _{q}^{n}\times \mathbb {Z} _{q}$

SH.Keygen: sample $L+1$ vectors $s_{0},\dots ,s_{L}$ uniformly from $\mathbb {Z} _{q}^{n}$ and compute, for all $l\in [L]$, $0\leq I\leq j\leq n$ and $\tau \in \{0,\dots ,\lfloor \log {q}\rfloor \}$, the value

$\psi _{l,i,j,\tau }:=\left(a_{l,I,j,\tau },b_{l,I,j,\tau }:=\langle a_{l,I,j,\tau },s_{l}\rangle +2\cdot e_{l,I,j,\tau }+2^{\tau }\cdot s_{l-1}\cdot s_{l-1}[j]\right)\in \mathbb {Z} _{q}^{n}\times \mathbb {Z} _{q}$

Where $a_{l,I,j,\tau }$ and $e_{l,I,j,\tau }$ are chosen uniformly at from $\mathbb {Z} _{q}^{n}$ and from the distribution $\chi$, respectively.

Let $\Psi :=\{\psi _{l,i,j,\tau }\}_{l,I,j,\tau }$ be the set of all these values. Notice that these seem like encryptions of $2^{\tau }\cdot s_{l-1}[i]\cdot s_{l-1}[j]$ (mod $q$), except these “ciphertexts” can’t be decrypted since the underlying message $2^{\tau }\cdot s_{l-1}[i]\cdot s_{l-1}[j]$ is not a single bit value.

The key generation might seem rather strange now, but publishing these “encryptions” of the quadratic terms in the secret keys $s_{l}$ is very important and we will explain this when we describe homomorphic multiplication.

The key-generation algorithm proceeds to choose a uniformly random matrix $A$ from $\mathbb {Z} _{q}^{m\times n}$ and an $m$-dimensional vector $e$ from the distribution $\chi ^{m}$. Set $b:=As_{0}+2e$.

The algorithm outputs the secret key $sk=s_{L}$, the evaluation key $evk:=\Psi$ and the public key $pk:=(A,b)$.

Bootstrapping BTS into a Fully Homomorphic encryption scheme

Efficiency of the Scheme