BGV

In 2011, Brakerski, Gentry and Vaikuntanathan (BGV) published the paper in which they introduce a new (leveled) fully homomorphic encryption (FHE) that improves performance and bases security on weaker assumptions than schemes from the previous generation.

A central conceptual contribution of this work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), without Gentry’s bootstrapping procedure.

Until recently, the BGV scheme was considered to be the most efficient homomorphic encryption scheme when performing the same operations on multiple ciphertexts at once.

New noise management technique

@TODO

Leveled Fully Homomorphic Encryption

Most of the work done by the will focus on the construction of a leveled fully homomorphic scheme, in the sense that the parameters of the scheme depend (polynomially) on the depth of the circuits that the scheme is capable of evaluating.

Definition. We say that a family of homomorphic encryption schemes $\{{\mathcal {E}}^{(L)}:L\in \mathbb {Z} ^{+}\}$ is leveled fully homomorphic if, for all $L\in \mathbb {Z} ^{+}$, they all use the same decryption circuit, ${\mathcal {E}}^{(L)}$ compactly evaluates all circuits of depth at most $L$ (that use some specified complete set of gates), and the computational complexity of ${\mathcal {E}}^{(L)}$'s algorithms is polynomial (a fixed polynomial for all $L$) in the security parameter $L$, and the size of the circuit (in the case of the evaluation algorithm).

The construction: FHE without bootstrapping

The authors base the security of their scheme on the hardness of Ring-Learning with errors problems, a generalisation of the classical LWE problem.

Let $\lambda$ be a security parameter, representing $2^{\lambda }$ security against known attacks.

Let $R$ be a ring. For any integer $q$, we write $R_{q}$ for the quotient $R/qR$.

Let $q=q(\lambda )$ be an odd modulus and $\chi =\chi (\lambda )$ a ``noise" distribution over $R$. Let $N=N(\lambda )$ be an additional parameter of the system which is larger than $3\cdot \log {q}$.

Let us assume that the plaintext is $R_{2}=R/2R$.

• E.Setup($1^{\lambda },1^{\mu }$): Choose a $\mu$-bit modulus $q$ and choose the other parameters $d=d(\lambda ,\mu )$, and $N=\lceil 3\log q\rceil$, $\chi (\lambda ,\mu )$ appropriately to ensure that the scheme is based on a Ring-LWE instance that achieves $2^{\lambda }$ security against known attacks. Let $R=\mathbb {Z} [X]/(x^{d}+1)$ and let params = $(q,d,N,\chi )$.

• E.SecretKetGen(params): Draw $s'\leftarrow \chi$. Set $sk=s\leftarrow (1,s')\in R_{q}^{2}$.

• E.PublicKeyGen(params, sk): Recall that the secret key is $sk=s=(1,s')$. This algorithm generates a (column) vector $A'\leftarrow R_{q}^{N}$, uniformly and a vector $e\leftarrow \chi ^{N}$. The algorithm computes

$b=A's'+2e$. Then, set $A$ to be the $N\times 2$ matrix obtained by setting $b$ on the first column followed by the entries of $-A'$. We remark that $A\cdot s=2e$. The algorithm outputs the public key $pk=A$.

• E.Enc(params, pk,m): To encrypt a message $m\in R_{2}=R/2R$, set $m\leftarrow (m,0)\in R_{q}^{2}$, sample $r\leftarrow R_{2}^{N}$ uniformly and output the ciphertext $c\leftarrow m+A^{T}\cdot r\in R_{q}^{2}$.

• E.Dec(params, sk, c): Output $m\leftarrow [[<c,s>]_{q}]_{2}$. Where $[\cdot ]_{q}$ denotes reduction into the range $(-q/2,q/2]$.

Correctness is easy to verify, whereas we refer to the paper for details upon security.

Key switching (Dimension reduction)

Recall that in the above scheme, the decryption equation for a ciphertext $c$ that encrypts a message $m$ under the secret key $s$ can be written as $m=E.Dec(params,s,c)=[[L_{c}(s)]_{q}]_{2},$ where $L_{c}$ is a linear operator which depends on $c$. To be precise, the latter is just the inner product $L_{c}(s)=<c,s>$ between two vectors in $R_{q}^{2}$.

To understand the key switching procedures, we have to (at least briefly) describe the following subroutines first:

• BitDecomp($x\in R_{q}^{n},q$): decomposes $x$ into its bit representation, namely $x=\sum _{j=0}^{\lfloor \log q\rfloor }2^{j}u_{j},$ where all of the $u_{j}\in R_{2}^{n}$. The procedure outputs the vector

$(u_{0},u_{1},\dots ,u_{\lfloor \log q\rfloor })\in R_{2}^{n\cdot \lceil \log q\rceil }$

• Powersof2($x\in R_{q}^{n},q$): outputs the vector

$(x,2\cdot x,\dots ,2^{\lfloor \log q\rfloor }\cdot x)\in R_{q}^{n\cdot \lceil \log q\rceil }.$

It is easy to see that for vectors $c,s$ of the same length $n$, we have:

$<BitDecomp(c,q),Powersof2(s,q)>=<c,s>{\pmod {q}}$.

Now, key switching consists of two procedures:

• SwitchKeyGen($s_{1}\in R_{q}^{n_{1}},s_{2}\in R_{q}^{n_{2}}$):

• • Run $A\leftarrow E.PublicKeyGen(s_{2},N)$ for $N=n_{1}\cdot \lceil \log q\rceil$.

• • Set $B\leftarrow A+Powersof2(s_{1})$ (Add Powersof2($s_{1}$)), which is an element of $R_{q}^{N}$ to $A$'s first column. Output $\tau _{s_{1}\to s_{2}}=B$.

• SwitchKey($\tau _{s_{1}\to s_{2}},c_{1}$): Output $c_{2}=BitDecomp(c_{1})^{T}\cdot B\in R_{q}^{n_{2}}$.

Note that, in SwitchKeyGen, the matrix $A$ consists of encryptions of $0$ under the key $s_{2}$. Then, pieces of the key $s_{1}$ are added to these encryptions of $0$. Thus, in some sense, the matrix $B$ consists of encryptions of pieces of $s_{1}$ (in a certain format) under the key $s_{2}$.

The authors establish the following correctness result (See Lemma 3 in the paper):

$[[<c_{2},s_{2}>]_{q}]_{2}=[[<c_{1},s_{1}>]_{q}]_{2}$

$c_{2}$ is a valid encryption of $m$ under the key $s_{2}$, and one can see that the noise of $c_{2}$ is larger by a small additive factor.

Modulus switching

If $c$ is a valid encryption of $m$ under the key $s$, and $s$ is (what the authors call) a ``short vector" and $c'$ is a simple scaling of $c$, i.e. $c'$ is the $R$-vector closest to $(p/q)\cdot c$ under the additional constraint that

$c'=c{\pmod {2}}$,

it turns out that (under some conditions) the vector $c'$ is a valid encryption of $m$, under the secret key $s$ using the modulus $p$.

In other words, under some conditions, one can change the modulus $q$ in the decryption equation to another (maybe smaller) modulus $p$, while preserving the correctness of decryption under the same secret key $s$.

For an integer vector $x$ and integers $q>p>m$, the authors define

$x'\leftarrow \mathrm {Scale} (x,q,p,r)$

to be the $R$-vector closest to $(p/q)\cdot x$ which satisfies $x'=x{\pmod {r}}.$

For the details concerning the conditions on $p,q$ and the vector $s$ required for the preservation of decryption correctness, we refer to Lemma 4 and its Corollary 1 in [1].

The aforementioned results assert that assuming $p$ is smaller than $q$ and $s$ has coefficients that are small in relation to $q$ , this ``modulus swithcing" procedure permits the evaluator to reduce the magnitude of the noise without knowing the secret key. The evaluator only has to know a bound on $\vert \vert s\vert \vert$.

Leveled FHE

We will only present the version of the scheme which is based on Ring version of LWE (R-LWE) and we reffer to Section 3.4 in [1] for an analogous construction of a scheme based on the classical LWE.

In the scheme, the authors use a parameter $L$ indicating the number of levels of arithmetic circuit that the FHE scheme should be able of evaluating.

FHE without Bootstrapping:

• FHE.Setup($1^{\lambda },1^{L}$): Takes as imput the security parameter and a number of levels $L$. Let $\mu :=\mu (\lambda ,L)=\theta (\log {\lambda }+\log {L})$ be a parameter (see the original paper [1] for details about its choice).

For $j=L$ to $0$ (output level), run $params_{j}\leftarrow E.Setup(1^{\lambda },1^{(j+1)\mu })$ to obtain a ladder of decreasing moduli from $q_{L}$ - which has $(L+1)\cdot \mu$ bits - down to $q_{0}=\mu$ bits.

For $j=L-1$ to $0$, replace the value of $d_{j}$ in $params_{j}$ with $d=d_{L}$ and the distribution $\chi _{j}$ with $\chi _{L}$. By this, one ensures that the ring dimension and the noise distribution do not depend on the circuit level, but the vector dimension $n_{j}$ might depend.

• FHE. KeyGen($\{params_{j}:j={\overline {0,L}}\}$ ): For $j=L$ down to $0$, do the following:

1. Run $s_{j}\leftarrow$ E.SecretKeygen($params_{j}$) and $A_{j}\leftarrow$ E.PublicKeyGen($params_{j},s_{j}$).

2. Set $s_{j}'\leftarrow s_{j}\otimes s_{j}$, namely $s_{j}$ tensored with itself.

3. Set $s_{j}''\leftarrow$ BitDecomp($(s_{j}',q_{j})$).

4. Run $\tau _{s_{j+1}''\to s_{j}}\leftarrow$ SwitchKeyGen($s_{j}'',s_{j-1}$). Omit this step when $j=L$.

• FHE.Enc($params,pk,m$): Take a message in $R_{2}$ and run $E.Enc(A_{L},m)$.

• FHE.Dec($params,sk,c$): Suppose the ciphertext is encrypted under the key $s_{j}$. Run $E.Dec(s_{j},c)$. The authors mention (and in fact this is the case for the implementation), that the ciphertext could be augmented with an index indicating which level it belongs to.

• FHE.Add($pk,c_{1},c_{2}$): The ciphertexts $c_{1},c_{2}$ must be encrypted under the same $s_{j}$. If they are not, one uses FHE.Refresh detailed below to make it so. Set $c_{3}\leftarrow c_{1}+c_{2}{\pmod {q_{j}}}$.

One should interpret $c_{3}$ as a ciphertext under $s_{j}'=s_{j}\otimes s_{j}$. Recall that the first coefficient of $s_{j}$ is equal to $1$.

Output

$c_{4}:=FHE.Refresh(c_{3},\tau _{s_{j}''\to s_{j-1}},q_{j},q_{j-1}).$

• FHE. Mult($pk,c_{1},c_{2}$): As before, the ciphertexts must be encrypted under the same $s_{j}$. If they are not, one uses FHE.Refresh to make it so.

First, multiply: the new ciphertext, under the secret key $s_{j}'=s_{j}\otimes s_{j}$, is the coefficient vector $c_{3}$ of the linear equation

$L_{c_{1},c_{2}}^{long}(x\otimes x).$

Remember one can see $L_{c_{1}}(x)\cdot L_{c_{2}}(x)=<c_{1},x>\cdot <c_{2},x>$ as a linear equation over the coefficients of $x\otimes x$, the tensoring of $x$ with itself. This is the meaning of the linear functional $L_{c_{1},c_{2}}$.