GSW - Bootstrapping

Gentry's bootstrapping theorem allows for converting a “somewhat homomorphic” encryption scheme (which supports only a bounded number of homomorphic operations) into a fully homomorphic encryption one (which has no such bound). The bounded nature of all known somewhat homomorphic schemes cannot be avoided due to “error” terms in their ciphertexts, which are necessary for security. The error grows as a result of performing homomorphic operations, and if it grows too large, the ciphertext will no longer decrypt correctly.

Bootstrapping the error of a ciphertext so that it can support more homomorphic operations, by homomorphically evaluating the decryption function on the ciphertext. The result is a ciphertext that still encrypts the original encrypted message. If the error coming from the homomorphic evaluation is smaller than the error in the original ciphertext, we say that the ciphertext is “refreshed”. To date, bootstrapping is the only known way of obtaining an unbounded FHE scheme, i.e., one that can homomorphically evaluate any efficient function using keys and ciphertexts of a fixed size.

Here we present an efficient bootstrapping method for a variant of the GSW scheme, as presented in the paper of Alperin-Sheriff and Peikert .

A "simpler" variant of the GSW cryptosystem

The authors present a variant of the GSW scheme which permits a tighter analysis of its error growth under homomorphic operations.

Given a modulus $q$, let us denote by $l=\lceil \log _{2}(q)\rceil$ and define the "gadget" (we will think of it as column) vector

${\mathfrak {g}}=(1,2,4,\dots ,2^{l-1})\in \mathbb {Z} _{q}^{l}.$

Remark that $2^{l-2}\in [q/4,q/2){\pmod {q}}$, according to our choice of $l$.

A randomized decomposition function. There is a randomized, efficiently computable function ${\mathfrak {g}}^{-1}:\mathbb {Z} _{q}\to \mathbb {Z} ^{l}$ such that $x\leftarrow {\mathfrak {g}}^{-1}(a)$ is "subgaussian" with parameter $O(1)$, and always satisfies $\langle g,x\rangle =a$.

In particular $x$ is randomized and has low entries with very large probability.

For vectors and matrices over $\mathbb {Z} _{q}$, define the randomized function $G^{-1}:\mathbb {Z} _{q}^{n\times m}\to \mathbb {Z} ^{nl\times m}$ by applying ${\mathfrak {g}}$ independently to each entry. Notice that for any $A\in \mathbb {Z} _{q}^{n\times m}$, if $X\leftarrow G^{-1}(A)$, then $X$ has small entries (is "subgaussian") and

$G\cdot X=A$, where $G={\mathfrak {g}}^{t}\otimes I_{n}=\mathrm {diag} ({\mathfrak {g}}^{t},\dots ,{\mathfrak {g}}^{t})\in \mathbb {Z} _{q}^{n\times nl},$

is the block matrix with $n$ copies of ${\mathfrak {g}}^{t}$ as diagonal blocks, and zeros elsewhere.

This version of the GSW scheme has as parameters a dimension $n$, a modulus $q$ and $l=\lceil \log _{2}(q)\rceil$, as defined above. For bootstrapping, we only work with ciphertexts encrypting messages in $\{0,1\}\subseteq \mathbb {Z}$. The ciphertext space is ${\mathcal {C}}=\mathbb {Z} _{q}^{n\times nl}$. For simplicity, we present just a symmetric-key scheme, which can be converted to a public key setting, similar to the one described in GSW.

A substantial difference between the original GSW scheme and this variant is the following: the homomorphic multiplication procedure in the current variant uses the randomized $G^{-1}(\cdot )$ operation presented above. This gives important advantages, such as very simple error analysis (using the subgaussianity property) and the ability to completely re-randomize the error in a ciphertext.

The algorithms of the scheme are as follows

• GSW.Gen(): choose ${\overline {s}}\leftarrow \chi ^{n-1}$ and output the secret key $s=({\overline {s}},1)\in \mathbb {Z} ^{n}$.

• GSW.Enc($({\overline {s}},1),\mu \in \mathbb {Z}$): choose ${\overline {C}}\leftarrow Z_{q}^{(n-1)\times nl}$ uniformly and $e\leftarrow \chi ^{nl}$, let $b^{t}=e^{t}-{\overline {s}}^{t}{\overline {C}}$ mod $q$, and output the ciphertext

$C=\left({\begin{array}{c}{\overline {C}}\\b^{t}\end{array}}\right)+\mu G\in {\mathcal {C}},$

where $G=\mathrm {diag} ({\mathfrak {g}}^{t},\dots ,{\mathfrak {g}}^{t})\in \mathbb {Z} _{q}^{n\times nl}$.

• GSW.Dec($s,C\in {\mathcal {C}}$): let $c$ be the penultimate column of $C$. Output $\mu =\lfloor \langle s,c\rangle \rceil _{2}$, where $\lfloor \cdot \rfloor _{2}:\mathbb {Z} _{q}\to \{0,1\}$ indicates whether its argument is closer modulo $q$ to $0$ or to $2^{l-2}$, the penultimate entry of ${\mathfrak {g}}$.

• GSW.Add($C_{1},C_{2}$): Output $C_{1}+C_{2}$.

• GSW.Multiply($C_{1},C_{2}$ ): Output $C_{1}\cdot G^{-1}(C_{2})$. This operation is a ramdomized one, as $G^{-1}$ is randomized. The multiplication is also right-associative.

The authors remark that a fresh ciphertext is just $\mu G$ plus a matrix which contains $nl$ independent LWE samples under the secret ${\overline {s}}$ which are pseudorandom if one assumes the hardness of $LWE_{n-1},q,\chi$.

Correctness and homomorphic operations

A ciphertext $C$ is said to be designed to encrypt a message $\mu \in \mathbb {Z}$ (under a secret key $s$) if it is a fresh encryption of $\mu$, or if $C$ is the sum (or the product) of two ciphertexts that are designed to encrypt $\mu _{1},\mu _{2}\in \mathbb {Z}$ and $\mu =\mu _{1}+\mu _{2}$.

The error vector of a ciphertext $C$ designed to encrypt a message $\mu$ under the key $s$ is $e^{t}=s^{t}C-\mu \cdot s^{t}G$ (mod $q$).

Notice that the matrix $\mu G$ is designed to encrypt $\mu$ and has error $0$.

Claim. If $C$ is designed to encrypt some $\mu \in \{0,1\}\subset \mathbb {Z}$ and has error vector $e^{t}$ whose penultimate coordinate has magnitude less than $q/8$, then GSW.Dec($s,C$) correctly outputs $\mu$.

This follows from the fact that $s=({\overline {s}},1)$ and the penultimate column of $G$ is $(0,\dots ,0,2^{l-2})$, where $2^{l-2}\in [q/4,q/2)$ mod $q$.

Let $C_{1},C_{2}$ be ciphertexts designed to encrypt $\mu _{1},\mu _{2}\in \mathbb {Z}$ and have error vectors $e_{1}^{t},e_{2}^{t}$. Then, their homomorphic sum has error vector $e_{1}^{t}+e_{2}^{t}$. Moreover, if $C_{1}\boxdot C_{2}$ is the homomorphic product of these ciphertexts, then for $X\leftarrow G^{-1}(C_{2})$ (subgaussian), we have the following

$s^{t}(C_{1}\boxdot C_{2})=s^{t}C_{1}\cdot X=(e_{1}^{t}+\mu _{1}\cdot s^{t}G)X$,

which as $G\cdot X=C_{2}$ is further equal to

$s^{t}(C_{1}\boxdot C_{2})=e_{1}^{t}X+\mu _{1}(e_{2}^{t}+\mu _{2}\cdot s^{t}G)=(e_{1}^{t}X+\mu _{1}e_{2}^{t})+\mu _{1}\mu _{2}\cdot s^{t}G.$