### Document

```Security Mediated
Certificateless Signatures
Without Pairing
Journal of Computers, Vol 5, No 12,
December 2010
Zhongmei Wan, Jian Weng, Jiguo Li
Citation: 3
Presenter:方竣民
2013/09/16
Outline
Introduction
 Preliminaries
 Construction of New Scheme
 Security Analysis
 Conclusions

Introduction

 Public key的管理需要大量的計算、儲


Shamir提出IBC(1984),直接對應user的
Email, ID
 User的private key由第三方KGC計算
 具有Key Escrow Problem

Introduction
Al-Riyami and Paterson提出CLPKC(2003)
 User的Partial Private Key由KGC利用
master secret key計算得到
 User自己產生一Secret Value及Public
Key, Private Key
 SMC密碼系統維護價值在於CLPKC
without 金鑰託管問題

Introduction

 以Random Oracle證明存在不可偽造性


 驗證需要3次

 該方案為第一個SMC簽章方法,基於困


Preliminaries
GCD( ,N) = 1 ,

for some integer
N=pq ,
, 其中p,q,p',q'均為質數
(N 為 內的平方子群)

Preliminaries
The Hardness Assumption of Factoring

Output: p 或 q

(對於一些安全參數k)
Construction of New Scheme
Setup
Input: 安全參數

1.隨機產生兩個質數

,

Construction of New Scheme
Setup
2.選擇
,合成Jacobi符號
3.計算
4.選擇4個one way function
KGC的master key:

Construction of New Scheme
KeyGeneration
User執行以下動作:
1.隨機產生兩個質數
,

2.隨機地選擇
User的private key
3.計算
User的public key

Construction of New Scheme
Register
User ID要註冊公鑰,KGC進行計算:

1.計算
2.計算
3.使得


Construction of New Scheme
Register
4.計算 為

5.寄出SEM private key

Construction of New Scheme
Signing

SEM確認該ID是否已經被revoke
If not, 雙方進行以下步驟:
1.SEM-Sign(I): SEM選擇

Construction of New Scheme
Signing
2.User-Sign(I): User選擇

3.SEM-Sign(II): SEM用中國餘式定理CRT

Construction of New Scheme
Signing
4.User-Sign(II): User確認

such that

such that

Construction of New Scheme
Signing
4.User-Sign(II):

such that

Construction of New Scheme
Verifying

1.計算
2.計算


Construction of New Scheme
Verifying
3.檢查等式

If equal then output "valid"
Else output "invalid"
Security Analysis
Theorem1

-存在不可偽造性

Security Analysis
Theorem1

Security Analysis
Theorem2

-存在不可偽造性

Security Analysis
Theorem2

Conclusions


SMC密碼系統解決了CLPKC的瞬時密

```