Publicly Verifiable Secret Sharing

In cryptography, a scheme is publicly verifiable (PVSS) if it is a scheme and if any party involved can verify the validity of the shares distributed by the dealer.

The method introduced here according to the paper by Chunming Tang, Dingyi Pei, Zhuo Liu, and Yong He is non-interactive and maintains this property throughout the protocol.

Contents

Initialization

The PVSS scheme dictates an initialization process in which:

  1. All system parameters are generated.
  2. Each participant must have a registered public key.

Excluding the initialization process, the PVSS consists of two phases:

Distribution

1.Distribution of secret s shares is performed by the dealer D, which does the following:

  • The dealer creates s_{1},s_{2}...s_{n} for each participant P_{1},P_{2}...P_{n} respectively.
  • The dealer publishes the encrypted share E_{i}(s_{i}) for each P_{i}.
  • The dealer also publishes a string mathrm{proof}_{D} to show that each E_{i} encrypts s_{i}

(note: mathrm{proof}_{D} guarantees that the reconstruction protocol will result in the same s.

2. Verification of the shares:

  • Anybody knowing the public keys for the encryption methods E_{i}, can verify the shares.
  • If one or more verifications fails the dealer fails and the protocol is aborted.

Reconstruction

1. Decryption of the shares:

  • The Participants P_{i} decrypts their share of the secret s_{i} using E_{i}(s_{i}).

(note: fault-tolerance can be allowed here: it’s not required that all participants succeed in decrypting E_{i}(s_{i}) as long as a qualified set of participants are successful to decrypt s_{i}).

  • The participant release s_{i} plus a string mathrm{proof}_{P_{i}} this shows the released share is correct.

2. Pooling the shares:

  • Using the strings mathrm{proof}_{P_{i}} to exclude the participants which are dishonest or failed to decrypt E_{i}(s_{i}).
  • Reconstruction s can be done from the shares of any qualified set of participants.

Chaums and Pedersen Scheme

A proposed protocol proving: log_{_{g1}}h_{1} = log_{_{g2}}h_{2} :

  1. The prover chooses a random rin boldsymbol{Zeta}_{q^*}
  2. The verifier send a random challenge c in _{R}boldsymbol{Zeta}_{q}
  3. The prover responds with s = r - c x(mathrm{mod},q)
  4. The verifier checks alpha_1 = g_{1}^s h_{1}^c and alpha_2 = g_{2}^s h_{2}^c

Denote this protocol as: mathrm{dleq}(g_1, h_1,g_2,h_2)
A generalization of mathrm{dleq}(g_1, h_1,g_2,h_2) is denoted as: text{dleq}(X, Y, g_1, h_1,g_2,h_2) where as: X = g_{1}^{x_1}g_{2}^{x_2} and Y = h_{1}^{x_1}h_{2}^{x_2}:

  1. The prover chooses a random r_1,r_2 in Z_{q}^* and sends t_1 = g_{1}^{r_1} g_{2}^{r_2} and t_2 = h_{1}^{r_1} h_{2}^{r_2}
  2. The verifier send a random challenge c in _{R}boldsymbol{Zeta}_{q} .
  3. The prover responds with s_1 = r_1 - cx_1 (mathrm{mod},q) , s_2 = r_2 - cx_2 (mathrm{mod},q) .
  4. The verifier checks t_1 = X^c g_{1}^{s_1}g_{2}^{s_2} and t_2 = Y^c h_{1}^{s_1}h_{2}^{s_2}

The Chaums and Pedersen method is an interactive method and needs some modification to be used in a non-interactive way: Replacing the randomly chosen c by a ‘secure hash’ function with m as input value.


Source

http://wikipedia.org/

See Also on BitcoinWiki