Abstract
Integer division is one of the most fundamental arithmetic operators and is ubiquitously used. However, the existing division protocols in secure multi-party computation (MPC) are inefficient and very complex, and this has been a barrier to applications of MPC such as secure machine learning. We already have some secure division protocols working in \(\mathbb {Z}_{2^n}\). However, these existing results have drawbacks that those protocols needed many communication rounds and needed to use bigger integers than in/output. In this paper, we improve a secure division protocol in two ways. First, we construct a new protocol using only the same size integers as in/output. Second, we build efficient constant-round building blocks used as subprotocols in the division protocol. With these two improvements, communication rounds of our division protocol are reduced to about 36% (87 rounds \(\rightarrow \) 31 rounds) for 64-bit integers in comparison with the most efficient previous one.
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Notes
- 1.
- 2.
- 3.
Linear operations are realized by computing the linear operations locally, and adding some constant \(a\) is realized by adding a share \((a,0)\).
- 4.
\(\widehat{\widehat{\cdot }}\) in step 9 means the decimal is multiplied by \(2^{2n'}\), instead of \(2^{n'}\).
- 5.
As a natural consequence of not expanding bit size, \(n'\) should be at most \(n\). Hence, we let \(n'\) be equal to \(n\) so that a rounding error is minimal.
- 6.
This means that if \(x[i]\ne 0\), then \(l_x\le i \le u_x\) (the same also holds for \(y\)). The converse is not assumed.
- 7.
Since we treat integers as elements of \(\mathbb {Z}_{2^n}\), in the case above, \((\lfloor \frac{N}{D} \rfloor +1)D\) is equal to \((\lfloor \frac{N}{D} \rfloor +1)D-2^n\) and less than \(N\).
- 8.
From the assumption that the exact quotient is in \(\{Q',Q'+1,\dots ,Q'+A-1\}\), \(N'\ge 0\) and \(N'< iD\) holds for some indexes \(i\).
- 9.
Note that \(\hat{\delta }\) depends only \(D\) in \(\mathsf {QGuess}\).
- 10.
Though we treat \(\mathsf {Pow}\) only over \(\mathbb {F}_p\), we can construct \(\mathsf {Pow}\) over \(\mathbb {Z}_{2^n}\) similarly.
- 11.
Matching with binary expression, the rightmost component of \(\mathbf{X} \) corresponds to \(\mathbf{X} [1]\).
- 12.
Note that a product of \(M\) numbers can be computed by executing a product of two numbers \(\lceil \log _2 M\rceil \) times.
- 13.
This setting was used in [18].
- 14.
Also, [19] constructed an exact division protocol in the semi-honest model, which is the same setting as our protocol.
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Acknowledgements
This work was partly supported by JST CREST JPMJCR19F6, the Ministry of Internal Affairs and Communications Grant Number 182103105 and JST CREST JPMJCR14D6.
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Hiwatashi, K., Ohata, S., Nuida, K. (2020). An Efficient Secure Division Protocol Using Approximate Multi-bit Product and New Constant-Round Building Blocks. In: Conti, M., Zhou, J., Casalicchio, E., Spognardi, A. (eds) Applied Cryptography and Network Security. ACNS 2020. Lecture Notes in Computer Science(), vol 12146. Springer, Cham. https://doi.org/10.1007/978-3-030-57808-4_18
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