1. Prepared by Dr. Lamiaa Elshenawy
Computer Security
Lecture 7
Ch.13
Digital Signatures
Prepared by Dr. Lamiaa Elshenawy

2. Digital Signatures ElGamal Digital Signature Scheme
Properties
Attacks and Forgeries
Digital Signature Requirements
Direct Digital Signature
ElGamal Digital Signature Scheme
Schnorr Digital Signature Scheme
Digital Signature Standard
The DSS Approach
The Digital Signature Algorithm

3. Digital Signatures Key Points
Digital signature authentication mechanism that enables the creator of a message to attach a code that acts as a signature.
Digital signature standard (DSS) NIST standard secure hash algorithm (SHA)
Encryption Algorithm
H(M)
Private Key
Digital Signature

4. Digital Signatures Key Points
Most important development of public-key cryptography
Digital Signature

5. Digital Signatures Generic Model

6. Digital Signatures Essential Elements

7. Digital Signatures Properties
Message Authentication

8. Digital Signatures Properties

9. Digital Signatures Attacks and Forgeries

10. Digital Signatures Attacks and Forgeries
Key-only attack: C A’s public key
Known message attack: C set of messages and their signatures.
Generic chosen message attack: C list of messages independent of A’s public key
Directed chosen message attack: C list of messages dependent of A’s public key signatures seen
know
access
choose
choose
before

11. Digital Signatures Attacks and Forgeries
Total break: C A’s private key
Universal forgery: C efficient signing algorithm that provides an equivalent way of constructing signatures on arbitrary messages
Selective forgery: C signature for chosen message
Existential forgery: C signature for at least one message. C control over the message
determine
find
forges
forges no

12. Digital Signatures Requirements
Signature bit pattern signed message
Signature information unique to the sender
forgery and denial
Easy digital signature
Easy copy of digital signature in storage
Infeasible computation digital signature
depends
use
prevent
produce
recognize & verify
retain
forge

13. Digital Signatures Direct Digital Signature
Source
Destination

14. Digital Signatures Schemes
ElGamal Digital Signature Scheme
“Taher AlGamal (1985)”
Schnorr Digital Signature Scheme
“Claus Peter Schnorr ( )”
Digital Signature Standard (DSS)

15. Digital Signatures ElGamal Scheme
Let q is prime number & α is a primitive root of q
Generate the private/ public keys
Sign the message

16. Digital Signatures ElGamal Scheme
Verify the message
If V1=V2 No
Not valid
Yes
Valid

17. Digital Signatures ElGamal Scheme
Let q=19;
Primitive roots of q= {2, 3, 10, 13, 14, 15};
α=10
Generate the private/ public keys
Alice wants to sign a message
Let m=4

18. Digital Signatures ElGamal Scheme
Verify the message

19. Digital Signatures ElGamal Scheme
Example
We consider q = 467; α = 2; XA = 127
Now YA is calculated: YA = αXA mod p = 2127 mod 467 = 132
So the Alice’s pair of keys is (127) , (467,2,132)
We take m= 100 and K = 213 for the signature of this message
Here we notice that (213, 466)= 1
Calculate mod 466 = 431
Having these parameters, we can start to calculate the signature of Alice on the message M, which is represented by the pair (S1,S2)
S1= αK mod q = 2213 mod 467 = 29
S2= K-1 [m−(XA S1)] mod q = 431( x 29) mod 466 = 51
Verification step:
αm mod q = (YA ) S1 (S1) S2 mod q
2100 ≡ 189 (mod 467)
13229 x 2951 ≡ 189 (mod 467)
The verification confirms that the signature is valid.

20. Digital Signatures Schnorr Scheme
Generate the private/ public keys
Sign the message

21. Digital Signatures Schnorr Scheme
Verify the message

22. Digital Signatures Schnorr Scheme
Generate the private/ public keys
Choose p = 23, q = 11, where 11 is a prime factor of 22 = 23-1.
Choose a such that a11 = 1 mod 23. Let a = 2, since 211 = 2048 = 1 mod 23.
Choose a random integer s, 0 < s < q. Let s= 9, since 9<11.
Generate a public key by calculating ν, where ν = 29 mod 23 =6
User’s Public key: ν = 6
User’s Private key: s = 9

23. Digital Signatures Schnorr Scheme
Sign the message
Customer chooses r = 3 < 11, and computes x = 23 mod = 8 Customer sends x = 8 to merchant
Merchant sends e=5 to customer
Customer calculates y = (3+9x5) mod 11 = 48 mod 11 = 4, and returns y = 4 to the merchant. The signature (e , y)
Verify the message
Merchant calculates x’ ν e mod p = 8 x 6 5 mod 23 = mod 23 = 16
Merchant also calculates a y mod p = 2 4 = 16
These are the same so the merchant accepts that the customer knows x

24. Digital Signatures Digital Signature Standard (DSS)
NIST Federal Information Processing Standard (FIPS 186)-DSS
DSS Secure Hash Algorithm (SHA)
new Digital Signature Algorithm (DSA)
DSS (1991) (1993,1996)
DSS (2000) (FIPS 186-2)
DSS (2009) (FIPS 186-3)
published
use
presented
proposed
revised
expand-version
presented
updated
presented

25. Digital Signatures Digital Signature Standard (DSS)
verify
random number

26. Digital Signatures Digital Signature Algorithm (DSA)

27. Digital Signatures Digital Signature Algorithm (DSA)

28. Digital Signatures Digital Signature Algorithm (DSA)
Let p, q, g are public to all
p= 18x71+1= prime number
q= 71 (prime divisor of (p-1))
g= 318 mod 1279=1157
Let x= 15, k=10
y= mod 71=851
User’s private key:{x=15}
User’s public key: {y=851}

29. Digital Signatures Digital Signature Algorithm (DSA)
Signing
r =( mod 1279) mod 71 = 32
s =[10 -1 (123+15x32)] mod 71=39
m=123, Signature={32,39}
Verifying
W=39-1 mod 71= 51
u1=123 x 51 mod 71=25
u2=32 x 51 mod 71=70
ν = [( ) mod 1279] mod 71=32
Test: ν=r the signature is valid

30. Thank you for your attention