CS4236 Tutorial 6

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CS4236 Tutorial 6

• Question 1 on Text Book pg 191
• Show that if someone discovers the value of k
  used in the ElGamal signature scheme, then a can
  also be determined.

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ElGamal Signature Scheme
Alice
Bob
Eve
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ElGamal Signature Scheme
download
Public to all (p,a,ß)
Alice
Bob
Signed and send to Bob (m, r, s)
listening
download
Eve
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Alice procedures

• ßaa ( mod p ), note that 1ltaltp-1 is secret
  number choose by Alice
• gcd(k,p-1) 1, note that k is another secret
  number choose by Alice.
• r ak ( mod p )
• s k-1 (m ar) (mod p-1)
• Send (m , r , s) to Bob

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Bob procedures

• Compute v1 ßr rs (mod p), v2 am
• If v1 v2 (mod p) then the signature is correct
  signed by Alice.

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Eve procedures

• Suppose Eve know k
• Since, s k-1 (m ar) (mod p-1),
• sk (m ar) (mod p-1)
• ar (m ks) (mod p-1)
• Note that in these steps, p, m, k, s, r, are
  known.
• By applying the extended Euclidean Algorithm, Eve
  get the possible answer of a. Note that the
  number of possible answer is gcd(r, p-1).
• Test which answer fulfils ß aa (mod p)
• a is found.