Perfect verification of modular scheme

G. V. Matveev ; V. V. Matulis

Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2018), pp. 4-9.

Résumé

Secret sharing schemes are used to distribute a secret value among a group of users so that only authorized set of them can reconstruct the original secret correctly. The modular secret sharing scheme (MSSS) we are studying is based on the Chinese Remainder Theorem. In this scheme the secrets $s(x), S(x), s_{1}(x),\dots , s_{k}(x)$ are defined as follows $s(x)=S(x) ~mod\, m(x), s_{i}(x)=S(x) ~mod\, m_{i}(x), i=1,2,\dots , k$. All the secrets and moduli are chosen from polynomial ring $F_{p}[x]$, and the reconstruction of secret $s(x)$ is carried out by applying the above-mentioned Chinese Remainder Theorem. The verification of any secret sharing scheme is understood as the protocol of verification by the participants of their partial secrets and (or) the protocol for verifying the legitimacy of the actions of the dealer. In this paper, we introduce a perfect verification protocol of MSSS. It means that none information leaks under distribution and verification. Two verification protocols are introduced in this paper. The first one is simpler and it depends on assumption about dealer honesty. If there is no such assumption verification is more complex. Both protocols are based on one work by J. Benalo and generalize the protocol proposed earlier by M. Vaskovsky and G. Matveev in two ways. First, the general, not only the threshold access structure is verified, and secondly, the dealer is not necessarily honest. Earlier, N. Shenets found the perfection condition of MSSS. Thus, if these conditions аre met, both the MSSS and its verification protocol are perfect.

Détail

Keywords: polynomial modular scheme, secret sharing, verification, secret, partial secret, finite field.

Bibliographie

[1] R. Cramer, I. Damgard, J. B. Nielsen, “Multiparty Computation from Threshold Homomorphic Encryption”, Advances in Cryptology – EUROCRYPT 2001 (Berlin), 2045 (2001), 280–300, Heidelberg: Springer | DOI | MR

[2] J. Bethencourt, A. Sahai, B. Waters, “Ciphertext-policy attribute-based encryption”, Proceedings of 2007 IEEE Symposium on Security and Privacy (Berkeley, California, USA), 2007, 321–334, Los Alamitos, California: IEEE Computer Society | DOI

[3] J. C. Benaloh, “Secret sharing homomorphisms: keeping shares of a secret secret (extended abstract)”, Advances in Cryptology – CRYPTO’86, 263 (1987), 251–260, Berlin, Heidelberg: Springer | DOI | MR

[4] T. V. Galibus, G. V. Matveev, “Verifikatsiya parametrov modulyarnogo razdeleniya sekreta”, Vestnik BGU. Fizika. Matematika. Informatika, 1 (2015), 76–79

[5] M. M. Vaskovskii, G. V. Matveev, “Verifikatsiya modulyarnogo razdeleniya sekreta”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 2 (2017), 17–22 | MR

[6] T. V. Galibus, “Verifikatsiya polinomialnogo modulyarnogo razdeleniya sekreta nad dvoichnym polem”, Vestnik Brestskogo gosudarstvennogo tekhnicheskogo universiteta. Fizika, matematika, informatika, 5 (2014), 26–27

[7] T. Galibus, G. Matveev, “Generalized Mignotte’s sequences over polynomial rings”, Electronic Notes Theoretical Computer Science, 186 (2007), 43–48 | DOI | MR | Zbl

[8] N. N. Shenets, “Ob informatsionnom urovne modulyarnykh skhem razdeleniya sekreta”, Doklady Natsionalnoi akademii nauk Belarusi. Seriya fiziko-matematicheskikh nauk, 54(6) (2010), 9–12 | MR | Zbl