Abstract
Matrix multiplication is one of the most basic and useful operations in statistical calculations and machine learning. When the matrices contain sensitive information and the computation has to be carried out in an insecure environment, such as a cloud server, secure matrix multiplication computation (MMC) is required, so that the computation can be outsourced without information leakage. Dung et al. apply the Ring-LWE-based somewhat public key homomorphic encryption scheme to secure MMC [TMMP2016], whose packing method is an extension of Yasuda et al.’s methods [SCN2015 and ACISP2015] for secure inner product. In this study, we propose a new packing method for secure MMC from Ring-LWE-based secure inner product and show that ours is efficient and flexible.
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Notes
- 1.
When encode a rectangular matrix, add zero terms to last rows (or/and columns) if the row (or/and column) number is smaller than \( \beta \), e.g., \(m<k=\beta \),
$$ \begin{array}{l} \widetilde{A}[rows](x) =A_1^{(r)} X\,+\,...\,+\,x^{(i-1)\beta }A_i^{(r)} X\,+\,...\,+\,x^{(m-1)\beta }A_{m}^{(r)} X\\ \widetilde{A}[columns](x) =A_1^{(c)} X'\,+\,...\,+\,x^{(j-1)\beta ^2}A_j^{(c)} X'\,+\,...\,+\,x^{(\beta -1)\beta ^2}A_\beta ^{(c)} X', \end{array} $$where \(X' = (1, x,..., x^{m-1})^T\).
- 2.
- 3.
References
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Acknowledgement
This work was partially supported by JSPS KAKENHI Grant Number JP15K00028 and JST CREST Number JPMJCR168A. We thank Takuya Hayashi and Mishra Pradeep Kumar for the useful discussion. We also greatly appreciate the anonymous reviewers for their thoughtful comments that helped improving the manuscript.
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A Correctness of Theorem 2
A Correctness of Theorem 2
Apart from Notation list-1.\(\sim \)3., the following list of notation should be useful:
Notation List-4. For a vector \(V=(v_1, ... , v_{\gamma })\), \({\mathsf { pol}}(V)= v_1x^{\delta _1}+ ... + v_{\gamma }x^{\delta _{\gamma }}\), let
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When V is a row or column vector of a matrix of size \(\beta \times \beta \), \(\gamma = \beta \).
Proof
According to Eqs. (6) and (5), we have
and
We should prove for any \(i,j=1,...,\beta \), \({\mathsf { pol}}(A_i^{(r)})\) in \(x^{2j {\beta }^2-\beta }\widehat{A}(x)\) and \({\mathsf { pol}}(B_j^{(c)})\) in \(x^{(i-1) \beta }\widehat{B}(x)\) satisfy exactly
Case 1: when \(j=1\), i.e., \(\langle {\mathsf { Vec}}(x^{2 {\beta }^2-\beta }\widehat{A}(x), {\mathsf { Vec}}(x^{(i-1) \beta }\widehat{B}(x)) \rangle = \langle {A}_i, {B}_1 \rangle \). It can be easily check that
\({\mathsf { deg}}({\mathsf { pol}}(A_i^{(r)}[k]))= {\mathsf { deg}}({\mathsf { pol}}(B_1^{(c)}[k])) = 2\beta ^2+(i-2)\beta + (k -1)\) for \(k=1,..., \beta \)
Case 2: when \(j \ge 2\), i.e., \(\langle {\mathsf { Vec}}(x^{2j {\beta }^2-\beta }\widehat{A}(x)), {\mathsf { Vec}}(x^{(i-1) \beta }\widehat{B}(x)) \rangle = \langle {A}_i, {B}_j \rangle \). It can be easily check that
\({\mathsf { deg}}({\mathsf { pol}}(A_i^{(r)}[k]))= {\mathsf { deg}}({\mathsf { pol}}(B_j^{(c)}[k])) = 2j\beta ^2+(i-2)\beta + (k -1)\), for \(k=1,..., \beta \)
Note. Since
we have
Therefore, our packing method works if
The proof for correctness of the MMC AB is complete. Correctness of the MMC BA can be proved similarly. \(\square \)
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Wang, L., Aono, Y., Phong, L.T. (2018). A New Secure Matrix Multiplication from Ring-LWE. In: Capkun, S., Chow, S. (eds) Cryptology and Network Security. CANS 2017. Lecture Notes in Computer Science(), vol 11261. Springer, Cham. https://doi.org/10.1007/978-3-030-02641-7_5
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