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Reusable fuzzy extractor from the decisional Diffie–Hellman assumption

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Abstract

Fuzzy extractors convert noisy non-uniform readings of secret sources into reliably reproducible, uniformly random strings, which in turn are used in cryptographic applications. Reusable fuzzy extractor allows multiple uses of the same secret source. In this paper, we construct the first strongly reusable fuzzy extractor which tolerates linear fraction of errors, with security tightly reduced to the decisional Diffie–Hellman (DDH) assumption in the standard model. Our construction is simple and efficient. Only two group operations and an evaluation of a hash function are added compared with the traditional construction of non-reusable fuzzy extractors.

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Notes

  1. The “strong” reusability in [2] is weaker than ours, since \(\varDelta w_{i,j}\) is designated by a PPT adversary.

  2. One can always translate a binary string to an element in \({\mathbb {Z}}_p^l\) for a proper l.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (NSFC No. 61672346).

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yunhua Wen, Shengli Liu & Shuai Han

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Yunhua Wen, Shengli Liu & Shuai Han

  3. Westone Cryptologic Research Center, Beijing, 100070, China

    Shengli Liu

Authors
  1. Yunhua Wen
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  2. Shengli Liu
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Correspondence to Shengli Liu.

Additional information

Communicated by J. H. Cheon.

Appendices

Syndrome based construction of secure sketch

An efficient \([n,k,2t+1]\)-linear error correcting code \(\mathcal {C}\) over \(\{0,1\}^n\) is a subspace of \(\{0,1\}^n\). Fix any \((n-k)\times n\) matrix H whose rows generate the orthogonal space \(\mathcal {C}^{\bot }\) as parity-check matirx of \(\mathcal {C}\). For any \(v\in \{0,1\}^n\), define syndrome \(\mathsf {syn}(v)=Hv\). Then \(v\in \mathcal {C}\Longleftrightarrow \mathsf {syn}(v)=0\). For any \(c\in \mathcal {C}\), \(\mathsf {syn}(c+e)=\mathsf {syn}(c)+\mathsf {syn}(e)=\mathsf {syn}(e)\). The syndrome captures all the information necessary for decoding.

The syndrome based construction of secure sketch [19] is given below:

  • \(\mathsf {SS}(w) = \mathsf {syn}(w)=s\).

  • \(\mathsf {Rec}(w',s)=w'-e\), where e is the unique vector of Hamming weight less than t such that \(\mathsf {syn}(e) = \mathsf {syn}(w')-s\).

Clearly, this secure sketch is deterministic with linear property.

Toeplitz matrix

In linear algebra, a Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. Any matrix of the form is a Toeplitz matrix:

$$\begin{aligned} \begin{pmatrix} a_0&{}\quad a_{-1}&{}\quad a_{-2}&{}\quad \cdots &{}\quad \cdots &{}\quad a_{-n+1}\\ a_1&{}\quad a_0&{}\quad a_{-1}&{}\quad \ddots &{}&{}\quad \vdots \\ a_2&{}\quad a_1&{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \vdots \\ \vdots &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad a_{-1}&{}a_{-2}\\ \vdots &{}&{}\quad \ddots &{}\quad a_1&{}\quad a_0&{}\quad a_{-1}\\ a_{n-1}&{}\quad \cdots &{}\quad \cdots &{}\quad a_2&{}\quad a_1&{}\quad a_0 \end{pmatrix} \end{aligned}$$
(15)

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Wen, Y., Liu, S. & Han, S. Reusable fuzzy extractor from the decisional Diffie–Hellman assumption. Des. Codes Cryptogr. 86, 2495–2512 (2018). https://doi.org/10.1007/s10623-018-0459-4

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