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Better Bootstrapping for Approximate Homomorphic Encryption

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Topics in Cryptology – CT-RSA 2020 (CT-RSA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12006))

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Abstract

After Cheon et al. (Asiacrypt’ 17) proposed an approximate homomorphic encryption scheme, HEAAN, for operations between encrypted real (or complex) numbers, the scheme is widely used in a variety of fields with needs on privacy-preserving in data analysis. After that, a bootstrapping method for HEAAN is proposed by Cheon et al. (Eurocrypt’ 18) with modulus reduction being replaced by a sine function. In this paper, we generalize the Full-RNS variant of HEAAN proposed by Cheon et al. (SAC, 19) to reduce the number of temporary moduli used in key-switching. As a result, our scheme can support more depth computations without bootstrapping while ensuring the same level of security.

We also propose a new polynomial approximation method to evaluate a sine function in an encrypted state, which is specialized for the bootstrapping for HEAAN. Our method considers a ratio between the size of a plaintext and the size of a ciphertext modulus. Consequently, it requires a smaller number of non-scalar multiplications, which is about half of the Chebyshev method.

With our variant of the Full-RNS scheme and a new sine evaluation method, we firstly implement bootstrapping for a Full-RNS variant of approximate homomorphic encryption scheme. Our method enables bootstrapping for a plaintext in the space \({\mathbb {C}}^{16384}\) to be completed in 52 s while preserving 11 bit precision of each slot.

K. Han—This work was done when the first author was in Seoul National University (SNU).

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Notes

  1. 1.

    After version 3.2, they use one temporary modulus instead of bit-decomposition as in [17].

  2. 2.

    In practice, \(p_i\)’s are chosen to have maximum sizes within the word size (<64 bits). On the other hand, sizes of \(q_j\)’s are depend on the precision of applications, and usually they are 40–45 bits.

  3. 3.

    In the case of SEAL v3.2, they use the bit-decomposition technique with the RNS-decomposition to reduce the noise growth. But, this method also has a drawback. It increases the length of the public key vector for key-switching further, which is directly related to the complexity of the process.

  4. 4.

    In Step 1, inverse NTT transform is needed for the next step (modulus raising).

  5. 5.

    Here, SEAL v.3.3 and HEAAN-RNS indicate the scheme corresponding to each paper and library.

  6. 6.

    Previous method uses a sine function and double angle formula for a sine function needs both \(\cos (t)\) and \(\sin (t)\) to compute \(\sin (2t)\).

  7. 7.

    The code for finding an approximate polynomial for the cosine function can be found at [15].

  8. 8.

    In fact, they use the nodes \(t_i = K \cos \left( {i\pi /n}\right) \) for \(0 \le i \le n\) instead of nodes \(t_i = K \cos \left( (2i-1)\pi / (2n+2) \right) \) for \(1 \le i \le n+1\). But, there is no big difference.

  9. 9.

    Here, SEAL v.3.3 and HEAAN-RNS indicate the schemes corresponding to each library and paper.

  10. 10.

    \(|t-\sin {t}|<O(t^3)\) for t near the origin.

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Correspondence to Kyoohyung Han .

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A Correctness and Noise Growth of Homomorphic Multiplication

A Correctness and Noise Growth of Homomorphic Multiplication

Before proving the correctness of the homomorphic multiplication, first remind the properties of \(\texttt {ModUp} \) and \(\texttt {ModDown} \) with the following three equations:

$$\begin{aligned}&\left\Vert \texttt {CRT} _{{\mathcal {C}}\cup {\mathcal {B}}}(\texttt {ModUp} ([a(x)]_{q_0},[a(x)]_{q_1},\dots ,[a(x)]_{q_\ell }))\right\Vert _\infty \le (\ell + 1) \cdot Q, \end{aligned}$$
(A.1)
$$\begin{aligned} \texttt {CRT} _{{\mathcal {C}}\cup {\mathcal {B}}}(\texttt {ModUp} ([a(x)]_{q_0},[a(x)]_{q_1},\dots ,[a(x)]_{q_\ell })) \equiv a(x) \bmod Q, \end{aligned}$$
(A.2)
$$\begin{aligned} \left\Vert \texttt {CRT} _{{\mathcal {C}}}(\texttt {ModDown} ([a(x)]_{q_0},\dots ,[a(x)]_{q_\ell },[a(x)]_{p_0},\dots ,[a(x)]_{p_{k-1}})) - \left\lfloor \dfrac{a(x)}{P} \right\rceil \right\Vert _\infty < k, \end{aligned}$$
(A.3)

where \({\mathcal {B}}=\{p_0,\dots ,p_{k-1}\}\) and \({\mathcal {C}}=\{q_0,\dots ,q_\ell \}\). With the above three equations and properties of RNS-Decompose and RNS-Power, we can prove the correctness of the homomorphic multiplication in our scheme.

Theorem 2

The algorithm such that

$$ b_3(x)+a_3(x)\cdot s(x) = M_1(x) \cdot M_2(x) + M_0(x) \cdot e_1(x) + M_1(x) \cdot e_0(x) + \epsilon (x), $$

where . Here, \(\textsf {ct} _i=(b_i(x),a_i(x)) \in {R}_Q^2\), and \(b_i(x)+a_i(x)\cdot s(x) = M_i(x) + e_i(x)\) for \(i = 0, 1\).

Proof

For simplicity, we assume that \(\ell = L\) and \((\ell +1)\) is a multiple of \(\alpha \). First, a vector \((d_0(x),d_1(x),d_2(x))\) which satisfies

$$\begin{aligned}&d_0(x) + d_1(x) \cdot s(x) + d_2(x) \cdot s(x)^2 = (M_0(x)+e_0(x)) \cdot (M_1(x)+e_0(x))\\&= M_0(x) \cdot M_1(x) + M_0(x) \cdot e_1(x) + M_1(x) \cdot e_0(x) + e_0(x) \cdot e_1(x)\\&= M_0(x) \cdot M_1(x) + e_2(x) \in {R}_Q. \end{aligned}$$

is obtained after Step 1.

In Step 2, since \(\ell = L\) and \((L+1)\) is a multiple of \(\alpha \), \(\beta \) equals to \({\texttt {\textit{dnum}}}\) and the zero-padding part can be omitted. Then,

$$ ([d_2(x)\cdot \hat{Q}_0^{-1}]_{Q_0}, \dots , [d_2(x)\cdot \hat{Q}_{{\texttt {\textit{dnum}}}-1}^{-1}]_{Q_{{\texttt {\textit{dnum}}}-1}}) = {\texttt {\textit{RNS-Decomp}}}_{{\mathcal {C}}'}(d_2(x)) $$

is returned after RNS-Decompose step.

Also, Modulus-Raise step returns vectors of length \(k+\ell +1\),

$$\begin{aligned}&([\tilde{d}^{(i)}_2(x)]_{q_0},\dots ,[\tilde{d}^{(i)}_2(x)]_{q_\ell },[\tilde{d}^{(i)}_2(x)]_{p_0},\dots ,[\tilde{d}^{(i)}_2(x)]_{p_{k-1}})\\&=\texttt {ModUp} _{{\mathcal {C}}_i\rightarrow {\mathcal {C}}\cup {\mathcal {B}}}([d_2(x) \cdot \hat{Q}_{i}^{-1}]_{q_{i\alpha }},\dots ,[d_2(x) \cdot \hat{Q}_{i}^{-1}]_{q_{(i+1)\alpha -1}}), \end{aligned}$$

where \(\tilde{d}^{(i)}_2(x) \in {R}_{PQ}\), for \(0\le i < {\texttt {\textit{dnum}}}\). From Eqs. A.1A.2, we can check that \(\tilde{d}_2(x)\) satisfies the following equations:

$$\begin{aligned} \tilde{d}^{(i)}_2(x) \equiv d_2(x) \cdot \hat{Q}_{i}^{-1} \bmod Q_i \text { and } \left\Vert \tilde{d}^{(i)}_2(x)\right\Vert _\infty \le (\alpha + 1) \cdot Q_i. \end{aligned}$$
(A.4)

Note that the norm of \(\tilde{d_2}^{(i)}(x)\) is still much smaller than PQ, and for this reason, \(\texttt {ModUp} \) does not harm the functionality of \({\texttt {\textit{RNS-Decompose}}}\) and \({\texttt {\textit{RNS-Power}}}\).

Next, we suppose that evaluation keys which satisfy \(B_i(x) + A_i(x) \cdot s(x) = P \cdot \hat{Q}_i \cdot s^2(x) + E_i(x) \in {R}_{PQ}\), where \(\left\Vert E_i(x)\right\Vert _\infty < e_\texttt {fresh}\), are generated in the key generation step. Then, the inner product step returns \((B'(x),A'(x)) = \sum _{i=0}^{\beta -1}\left[ \tilde{d_2}^{(i)}(x) \cdot (B_i(x),A_i(x)) \right] \) and it satisfies the following equation:

$$\begin{aligned} B'(x)+A'(x) \cdot s(x)&= P \sum _{i=0}^{\beta -1}\left( \tilde{d_2}^{(i)}(x) \cdot \hat{Q}_i \cdot s^2(x)\right) + \sum _{i=0}^{\beta -1}\left( \tilde{d_2}^{(i)}(x) \cdot E_i(x)\right) \\&= P \cdot d_2(x) \cdot s^2(x) + E'(x) \in {R}_{PQ}, \end{aligned}$$

where and N is the dimension of the ring.

After that, we apply modulus-down process to revert the modulus space from \({R}_{PQ}\) to \({R}_Q\) and to reduce the size of \(E'(x)\). Let \((\tilde{B}(x),\tilde{A}(x))\) be the return of modulus-down step with CRT decomposed representation. From the modulus switching technique and Equation A.3, we can see that \((\tilde{B}(x),\tilde{A}(x))\) has the following property:

$$\begin{aligned} \tilde{B}(x)+\tilde{A}(x)\cdot s(x)&= d_2(x) \cdot s^2(x) + \left\lfloor \dfrac{E'(x)}{P} \right\rceil + \epsilon (x) \in {R}_Q, \end{aligned}$$

where \(\left\Vert \epsilon (x)\right\Vert _\infty < \left\Vert s(x)\right\Vert _1\). Since , each coefficient of \({E'(x)}/{P}\) is in the range \((-0.5,0.5)\), and thus rounding of the polynomial becomes a zero polynomial. Therefore, it follows that \(\tilde{B}(x)+\tilde{A}(x)\cdot s(x) = d_2(x)\cdot s(x)^2 + \epsilon (x) \in {R}_Q\).

At the last step, we compute and return \((b_3(x),a_3(x)) = (d_0(x)+\tilde{B}(x),d_1(x)+\tilde{A}(x))\). Then, from the equation

$$\begin{aligned} b_3(x) + a_3(x) \cdot s(x)&= d_0(x) + d_1(x) \cdot s(x) + d_2(x) \cdot s(x)^2 + \epsilon (x) \\&= M_0(x) \cdot M_1(x) + e_2(x) + \epsilon (x), \end{aligned}$$

the correctness of homomorphic multiplication is followed. Furthermore, the size of the noise after multiplication is given by \(M_0(x)\cdot e_1(x) + M_1(x)\cdot e_0(x) + e_0(x) \cdot e_1(x) + \epsilon (x)\), where \(\left\Vert \epsilon (x)\right\Vert _\infty < \left\Vert s(x)\right\Vert _1\).    \(\square \)

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Han, K., Ki, D. (2020). Better Bootstrapping for Approximate Homomorphic Encryption. In: Jarecki, S. (eds) Topics in Cryptology – CT-RSA 2020. CT-RSA 2020. Lecture Notes in Computer Science(), vol 12006. Springer, Cham. https://doi.org/10.1007/978-3-030-40186-3_16

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