Geodesic
Faulty share detection in Shamir’s secret sharing
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 274-282 Cet article a éte moissonné depuis la source Math-Net.Ru

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For Shamir's secret key sharing algorithm, we develop the procedure for detection of faulty shares. This procedure consists of the error locator polynomial construction for the data set $ \{ (x_j,y_j)\}_{j=1}^N $ with $ y $ values generated from $ x $ ones by a polynomial interpolant of a degree $ n < N-1 $ with possible occurrence of some errors. The error locator polynomial is sought out in the form of an appropriate Hankel polynomial $$ \mathcal H_{L}(x;\{ \tau \}) := \left| \begin{array}{lllll} \tau_0 & \tau_1 & \tau_2 & \ldots & \tau_{L} \\ \tau_1 & \tau_2 & \tau_3 &\ldots & \tau_{L+1} \\ \vdots & \vdots & \vdots & & \vdots \\ \tau_{L-1} & \tau_{L} & \tau_{L+1} & \ldots & \tau_{2L-1} \\ 1 & x & x^2 & \ldots & x^{L} \end{array} \right| \, , $$ where $ \tau_{\ell} := \displaystyle \sum_{j=1}^{N} y_j \frac{x_j^{\ell}}{W^{\prime}(x_j)} $; $ \displaystyle W(x):=\prod_{j=1}^N (x- x_j) $.
Keywords: Shamir’s secret sharing, Hankel polynomials
Mots-clés : polynomial interpolation, error correction. 
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     author = {A. Yu. Uteshev and A. V. Marov},
     title = {Faulty share detection in {Shamir{\textquoteright}s} secret sharing},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {274--282},
     year = {2019},
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A. Yu. Uteshev; A. V. Marov. Faulty share detection in Shamir’s secret sharing. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 274-282. http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a9/

[1] Shamir A., “How to share a secret”, Communications of the ACM, 22:11 (1979), 612–613 | DOI | MR | Zbl

[2] Welch L. R., Berlekamp E. R., Error correction for algebraic block codes, US Patent No 4 633 47, Dec. 30, 1986 (accessed: 10.01.2019) https://patentscope.wipo.int/search/en/detail.jsf?docId=US37599078

[3] Uteshev A. Yu., Baravy I., Solution of interpolation problems via the Hankel polynomial construction, 2016, arXiv: (accessed: 10.01.2019) 1603.08752 [cs.SC] | Zbl

[4] Uteshev A. Yu., Baravy I., “Solution of the rational interpolation problem via the Hankel polynomial construction”, Vestnik of Saint Peterburgs University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2016, no. 4, 31–43 | MR