CSCI381 Fall 2005

1
Digital Signatures

• CSCI381 Fall 2005
• GWU

2
Definition

• P set of plaintext
• S set of signatures
• K keyspace
• private function sigk P ? S
• public function verK P X S ? true, false
• verK(m, s) true iff sigK(m) s else verK(m,
  s) false
• m, sigK(m) is a signed message

3
El Gamal Digital Signature

• For a key K (p, ?, ?, a) ? ?a mod p a
  private
• Choose random k invertible in Zp-1
• sigK(x, k) (??k mod p, ?(x-a?)k-1 mod p-1)
• verK(x, (G, D)) true ? ?GGD?x mod p
• Depends on security of the DL problem Find a
  given p, ?, ?

4
Digital Signature Example

• K (p11, ?2, ?5, a4) ? ?a mod p a private
• Choose random k3 invertible in Zp-1
• sigK(x7, k3) (??k mod p, ?(x-a?)k-1 mod
  p-1) (8, 5)
• verK(x, (G, D)) true ? ?GGD?x mod p (7 mod 11)

5
DSA

• Uses SHA(x) instead of X

6
Discrete Log in Elliptic Curves

• As before, only group is no longer Zp
• The problem is to determine a given P and Q aP
  in the elliptic curve group
• The best-known algorithm for breaking DL over Zp
  takes less time than that for breaking DL over an
  elliptic curve group of the same size