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Provably secure randomized blind signature scheme based on bilinear pairing

Source: Computers and Mathematics with Applications

Author: Chun-I Fan , Wei-Zhe Sun, Vincent Shi-Ming Huang

Presenter: 林志鴻

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Outline

Introduction Preliminaries Randomized blind signature Performance and security Analysis Conclusion

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Introduction

User Signer

+ 盲因子 =(1) (2) +

=

(3) －盲因子 =

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Introduction(cont.)

Usage of Blind Signature Anonymous electronic voting Untraceable electronic cash system

Security properties of Blind Signature Unlinkability Unforgeability randomization

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Unlinkability

Signer

A

B

A? or B?

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Blind signature with randomization

分成六個演算法 KeyGen(k) → (SK, PK) Blind(m, r, u) → α Sign(α,y, SK) → t Unblind(t, r) → s RandMix(u, y) → c ； σ=signature-message Verify(σ,PK) → {0,1}

Verify((Unblind (Sign (Blind (m, r, u),y,SK),r),m, RandMix(u, y) ),PK)=1

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Outline

Introduction Preliminaries Randomized blind signature Performance and security Analysis Conclusion

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Preliminaries

Bilinear Pairing GDH Groups

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Bilinear Pairing

e : G1 × G1 → G2

Bilinearity Non-degeneracy Computability

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GDH Groups

對於一個循環群 G CDH problem ︰

對 a,b Zq∈ 給定 (P,aP,bP) ∈ G 計算 abP DDH problem ︰

對 a,b,c Zq ∈ 給定 (P,aP,bP,cP) ∈ G 判斷 c=ab

若存在一多項式時間演算法 A 可解決 DDH問題但不存在任何演算法可解決 CDH 問題則此循環群 G 稱為 GDH Groups

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Outline

Introduction Preliminaries Randomized blind signature Performance and security Analysis Conclusion

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Randomized blind signature

Initialization phase Blinding phase Signing phase Unblinding phase Verification phase

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Randomized blind signature (cont.)

Initialization phase

1. 輸入秘密參數 k 產生兩個 order q 的循環群G1,G2 ,P 為 G1生成元 , e: G1× G1→G2

2. 簽章者選取兩個私鑰 x1,x2 Zq∈ * 產生相對應的公鑰 Pub1 = x1P, Pub2 = x2P ,H:{0,1}*→G1

*

params = (q, H,G1,G2,e,P, Pub1, Pub2)

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Randomized blind signature (cont.)

Blinding phase1. 當使用者發送簽章要求時，簽章者隨機選取

y Z∈ p* 傳送 ρ= yP 給使用者

2. 使用者準備明文 m 並隨機選取 u,r1,r2 Z∈ p* ，設

定隨機參數 C = u ρ

3. 計算盲訊息α1 = r1H(m || C) + r2Pα2 = r1u (mod q)

4. 傳送 (α1, α2 ) 給簽章者

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Randomized blind signature (cont.)

Signing phase 簽章者計算 T = x1α1 + x2yα2P並回傳給使用者

Unblinding phase使用者計算 S = r1

-1(T – r2Pub1)此時簽章 -訊息組為 (S,m,C)

Verification phase驗證式子 e(S, P) = e(H(m || C), Pub1)e(C, Pub2)

Pub1 = x1P, Pub2 = x2Pρ= yP ,C = u ρα1 = r1H(m || C) + r2Pα2 = r1u (mod q)

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Randomized blind signature (cont.)

整體流程

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Outline

Introduction Preliminaries Randomized blind signature Performance and security Analysis Conclusion

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Performance and security Analysis

[11]A. Boldyreva[12]H. Elkamchouchi, Y. Abouelseoud[13]Y. Yu, S. Zheng, Y. Yang[14] [15]F. Zhang, K. Kim

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Outline

Introduction Preliminaries Randomized blind signature Performance and security Analysis Conclusion

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Conclusion

本文提出了一個提供具有隨機屬性的 pairing-based 盲簽章並正式的證明此簽章具有 unlinkability, unforgeability, 和 randomization properties 。

本文提出的方法為第一個可證明安全的隨機化盲簽章