1. 第四章
數位簽章

2. 數位簽章 ‧RSA Digital Signature
(R. L. Rivest, A. Shamir, and L. M. Adleman, 1978)
‧ElGamal Digital Signature
(T. ElGamal, 1985)
‧Schnorr’s Digital Signature
(C. P. Schnorr, 1989)
‧Blind Signature
(D. Chaum, 1983)

3. Introduction Three Basic Services of Cryptography
1. Secrecy (provided by cryptosystems)
2. Authenticity (provided by digital signature scheme)
3. Integrity (provided by digital signature scheme)
Two Famous Digital Signature Schemes
1. RSA Digital signature scheme (based on the factorization problem)
2. ElGamal digital signature scheme (based on the discrete logarithm)

4. Introduction
Applications:
Blind signature (for electronic commerce)

5. Check Message=?Message
Introduction
The Model of Digital Signature
Signer’s public key
Signer’s secret key
Verification
Function
Sign Function
Message
Signature
Message
Message
Check Message=?Message

6. RSA Public Key Cryptosystem and Digital Signature Scheme
Rivest, Shamir, and Adleman proposed in 1978
RSA Public Key Cryptosystem
◆ Security Basis: Factorization Problem.
◆ Construction:
1. Choose two large prime numbers P and Q, then compute
N=P×Q.
2. Select an integer e such that gcd(e, (N))=1.
3. Compute d such that e×d mod (N)=1.
4. Public key = (N, e).
5. Private key = (P, Q, d).

7. RSA Public Key Cryptosystem and Digital Signature Scheme
RSA Digital Signature Scheme
Sign Function: Signature S=Md mod N.
Verification Function: M=Se mod N.
Example
P=11,Q=13, N=143, and (143)=120.
e=103, then d=7 (for 103×7 mod 120=1 ).
Sign for M=3: S=37 mod 143=42.
Verification: M= Se mod N = mod 143=3.

8. ElGamal Public Key Cryptosystem and Digital Signature Scheme
ElGamal proposed in 1985
ElGamal Public Key Cryptosystem
◆ Security Basis: Discrete Logarithm Problem
1. If P is a large prime and g and y are integers, find x such that y=gx mod P.
2. The security restriction on P: P-1 must contain a large prime factor Q.
◆ Construction:
1. Choose a large prime number P and a generator g of GF(P).
2. Private key: a random integer x between 1 and P-1.
3. Public key: y=gx mod P.

9. ElGamal Public Key Cryptosystem and Digital Signature Scheme
ElGamal Digital Signature Scheme
Sign Function: Signature (r, s) for message M.
1. Select a random integer k between 1 and P-1 such that gcd(k, P-1)=1.
2. Compute r=gk mod P.
3. Compute s=k-1(M-xr) mod (P-1).
Verification Function:
Verify by checking whether gM mod P = (rs) ×(yr) mod P.
(rs) ×(yr)=g(M-xr) × gxr = g(M-xr)+xr=gM mod P.

10. ElGamal Public Key Cryptosystem and Digital Signature Scheme
Example
P=23, g=5.
x=3, then y=10 (for 53 mod 23=10 ).
Sign for the message M=8.
Select k=5 between 1 and 22 (P-1).
Compute r = gk mod P = 55 mod 23 = 20.
Compute s = k-1(M-xr) mod (P-1) = 5-1(8-3×20) mod 22 = 9×14 mod 22 = 16.
Verification:
gM= 58 mod 23 =16
(rs)(yr) mod P = 2016 × 1020 mod 23= 13×3 mod 23 = 16.

11. Schnorr’s Digital Signature Scheme
Sign Function: Signature (r, s) for message M.
1. Select a random integer k between 1 and P-1.
2. Compute r = h(M, gk mod P).
3. Compute s = k + x*r mod (P-1).
, where the secret key x  the public key y= g-x mod P
4. Send (M, r, s) to the receiver.
Verification Function:
Compute gk mod P=gsyr mod P.
Verify by checking whether r = h(M, gk mod P).

12. Blind Signature D. Chaum proposed in 1983
D. Chaum’s Blind Signature Scheme
◆ It uses the RSA algorithm.
Security Basis: Factorization Problem
◆ Construction:
Bob has a public key, e, a private key, d, and a public modulus, N. Alice wants Bob to sign message M blindly.
1. Alice chooses a random integer k between 1 and N. Then she blinds M by computing t = Mke mod N.
2. Bob signs t, td=(Mke)d mod N.
3. Alice unblinds td by computing s=td/k mod N = Md mod N. s is the signature of message M.

13. Blind Signature Property: Untraceable
Applications: Blind signature can be used in electronic cash system.
signs coins
database
Bank
1. t=SN×ke mod N
SN: Serial #
k: random number
2. t
7. Coin
3. td mod N
Consumer
Merchant
5. Coin
4. s=(td)/k mod N=SNd mod N
6. Verify the signature s
Coin: SN＋s