Geodesic
Verification of modular secret sharing
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2017), pp. 17-22.

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In the present paper new scheme of secret verification are constructed. Verification with trusted party participation is conducted with help of an external device, which takes an arbitrary polynomial $S(x)$, input element $x_{0}\in F_{p^{n}}$ and returns a value $\xi S(x_{0})$ , where $\xi$ is an $F_{p^{n}}$ – valued uniformly distributed random variable. It is shown that using of such device allows any user to verify his secret. Polynomial verification scheme is based on verification of divisibility $g(x)|f(x)$ in the ring $Z(x)$. Only a value of polynomial $S(x)$ in unknown point $x=l$ is disclosed at the proposed verification method. Benaloh’s verification of the modular scheme allows any shareholder to ensure in consistency of all partial secrets, i. e. any legal group of shareholders can restore the secret $S(x)$ correctly. None information about the secret $S(x)$, excepting a prior information, is disclosed. The proposed protocols can be used safely for schemes over arbitrary finite fields without additional restrictions on a size of a filed.
Keywords: polynomial modular scheme; secret; partial secret; finite field. 
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M. M. Vas'kovskii; G. V. Matveev. Verification of modular secret sharing. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2017), pp. 17-22. http://geodesic.mathdoc.fr/item/BGUMI_2017_2_a2/

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