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SCIENCE CHINA Information Sciences, Volume 63 , Issue 3 : 130101(2020) https://doi.org/10.1007/s11432-018-9781-0

Blockchain-based multiple groups data sharing with anonymity and traceability

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  • ReceivedOct 31, 2018
  • AcceptedJan 23, 2019
  • PublishedAug 9, 2019

Abstract


Acknowledgment

This work was supported by National Cryptography Development Fund ( Grant No. MMJJ20180110).


References

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  • Figure 1

    (Color online) The system model of our scheme.

  • Figure 2

    The structure of the blocks.

  • Figure 3

    (Color online) (a) Time cost of a user and manager in initialization; (b) time cost of a user and manager in data sharing.

  • Figure 4

    (Color online) (a) The impact of data blocks size on the user audit overhead; (b) the impact of group members size on the user trace overhead.

  • Table 1   Comparison of the schemes in terms of the verification process
    Scheme [15] Scheme [18] Scheme [29] Scheme [23]Our scheme
    Communication $O(1)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$
    Server computation $O(t)$ $~tO({\rm~log}L)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$
    Verifier computation $O(t)$ $~tO({\rm~log}L)$$tO({\rm~log}L)$ $tO({\rm~log}L)$ $tO({\rm~log}L)$
  •   

    Algorithm 1 KeyGen()

    Require:Prime numbers $p,q$ with $l$ bits;

    Output:pk, sk;

    Compute $P=2p+1,~Q=2q+1$, let $n=PQ$;

    Select random elements $g,h,a_1,a_2,g_1,g_2,\delta_1,~\delta_2~\in_R~QR(n)$, where $QR(n)$ denotes the quadratic residue of group $\mathbb{Z}_{q}^{*}$;

    Select random $X_{j}~\in~\mathbb{Z}^{*}_{q}$, and compute $Y_{j}=g^{X_{j}},~j\in[1,N]$;

    Let ${\rm~pk}=\{Y_{j},g,h,a_1,a_2,g_1,g_2,\delta_1,~\delta_2,n\},~{\rm~sk}=\{X_j,~j\in~[1,N]\}$;

    return pk, sk;

  • Table 2   Functionality comparison of the schemes
    Scheme [15] Scheme [18] Scheme [29] Scheme[23]Our scheme
    Anonymity No No No Yes Yes
    TraceabilityNo No No YesYes
    Number of groups One OneOne One Multiple
    Public auditing No Yes Yes Yes Yes
    Non-frameability No No No NoYes
    Third-party auditor Yes YesNo Yes No
  •   

    Algorithm 2 GenSign()

    Require:Secret key $\{A_i,e_i\}$, system parameters $\{Y_j,g,h,a_1,a_2,g_1,g_2,\delta_1,~\delta_2,n\}$, and message $M$;

    Output:Signature $\sigma$ on message $M$;

    Select random number $r~\in~\{0,1\}^{2l}$, and compute the following values:$T_1=A_iY_{j}^{r}$, $T_2=g^r$, $T_3=g^{e_{i}}h^r$;

    Select random numbers:newline $r_1\in~\pm~\{0,1\}^{\kappa(\gamma_1+k)}$;newline $r_2\in~\pm~\{0,1\}^{\kappa(\lambda_2+k)}$;newline $r_3\in~\pm~\{0,1\}^{\kappa(\gamma_1+2l+k+1)}$;newline $r_4\in~\pm~\{0,1\}^{\kappa(2l+k)}$;newline then, compute the following values:newline $d_1=T_1^{r_1}/(a_1^{r_2}Y_{j}^{r_3})$;newline $d_2=T_2^{r_1}/(g^{r_3})$;newline $d_3=g^{r_4}$;newline $d_4=g^{r_1}h^{r_4}$;

    Generate a hash valuenewline $c=H_1(g\|h\|Y\|a_1\|a_2\|T_1\|T_2\|T_3\|d_1\|d_2\|d_3\|d_4\|M)$;

    Compute the following values:newline $s_1=r_1-c(e_i-2^{\gamma_1})$;newline $s_2=r_2-c(x_i-2^{\lambda_2})$;newline $s_3=r_3-cre_i$;newline $s_4=r_4-cr$;

    Generate the signature:newline $\sigma=(c,s_1,s_2,s_3,s_4,T_1,T_2,T_3)$;

    return $\sigma$.

  •   

    Algorithm 3 Ver()

    Require:System parameters $\{Y_{j},g,h,a_1,a_2,g_1,g_2,\delta_1,~\delta_2,n\}~$ and message $M$ with signature $\sigma$.

    Output:True or False;

    Compute the following values:newline $d'_1=a_1^{c}T_{1}^{s_1-c2^{\gamma_1}}/a_1^{s_2-c2^{\gamma_1}}Y_{j}^{s_3}$;newline $d'_2=T_2^{s_1-c2^{\gamma_1}}/g^{s_3}$;newline $d'_3=T_{2}^{c}g^{s_4}$;newline $d'_4=T_{3}^{c}g^{s_1-c2^{\gamma_1}}h^{s_4}$;

    Generate a hash valuenewline $c'=H_1(g\|h\|Y_{j}\|a_1\|a_2\|T_1\|T_2\|T_3\|d'_1\|d'_2\|d'_3\|d'_4\|M)$;

    if $c=c'$ then

    return `T';

    else

    return `F';

    end if

    return $\sigma$;