DLP Discrete Logarithm Problem

1
DLP (Discrete Logarithm Problem)

• Suppose p is an odd prime.
• Zp0,1,,p-1 is a finite field.
• Zp the set of integers which are relatively
  prime to p.
• a ? Zp gcd(a, p)11,,p-1
• it is a cyclic multiplicative group.
• ? is a generator of Zp ,
• i.e. , Zp ? 0 mod p, ? 1mod p, , ? p-2 mod
  p.
• DLP problem
• Given any a, compute b ? ? a (mod p) is easy.
• given any b, find an a such that b ? ? a (mod
  p) is difficult.
• Denoted as a log ? b. Omit mod p for
  simplicity.

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Example of DLP

• p17, Zp 1,2,,16. ? 3 a generator,
• Zp 30, 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ,
  39 , 310 , 311 , 312 , 313 , 314 ,315
• 1, 3, 9, 10, 13, 5, 15,11,16,14, 8, 7,
  4, 12, 2, 6
• Note 316 mod 17 1.

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Example of DLP (cont.)
Given a10, b ? 310 mod 17 ??.
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Given b14, what is a ??
By searching the table, a9.
When p is large, table is very large.
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ElGamal public key system

• Suppose p is a prime such that DLP in (Zp , ?)
  is infeasible and ? is a generator.
• K(p, ?, a, ?) where ? ? ?a mod p.
• publicize p, ?, ?, but keep a secret.
• For any x? Zp , select a random k ? Zp-1, define
  
• eK(x,k) (y1, y2) where
• y1 ? ? k mod p and y2 ? x? k mod p
• For y1, y2 ? Zp, define dK(y1, y2) ? y2 (y1a)-1
  mod p

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Example of ElGamal

• p17, ? 3, a6, ? ?15
• Message x11, then encryption is
• Randomly select secret k3.
• y1 ? ? k mod p ? 33 mod 17 10,
• y2 ? x? k ? 11?153 mod 17 11?9 14.
• So (10,14) is the encrypted message.
• Decryption
• y2 (y1a)-1
• 14 ?(106)-1 mod 17 14 ?9-1 mod 17 14 ? 2 mod
  17 11.

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Discussions of ElGamal

• Proof of ElGamal ???
• Why k ? Zp-1 ? also, k ? 0.
• Ciphertext doubles plaintext.
• Multiple ciphertexts (p-2) corresponding to one
  plaintext.

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ElGamal signature scheme

• Suppose p is a prime such that DLP in (Zp , ?)
  is infeasible and ? is a generator.
• K(p, ?, a, ?) where ? ? ?a mod p.
• publicize p, ?, ?, but keep a secret.
• For any x? Zp , select a random k ? Zp-1,
  define
• sigK(x,k) (?, ? ) where
• ? ? ? k mod p and ? ? (x-a ?)k -1 mod p-1
• For x, ? ? Zp and ? ? Zp-1,define
• verK(x, (?, ? )) true iff ? ? ? ? ? ? x mod p