General

1
Public Key Algorithms
CS 6262 Spring 03
2
Modular Arithmetic

• Public key algorithms are based on modular
  arithmetic.
• Modular addition.
• Modular multiplication.
• Modular exponentiation.

3
Modular Addition

• Addition modulo (mod) K
• Poor cipher with (dkdm) mod K, e.g., if K10 and
  dk is the key.
• Additive inverse addition mod K yields 0.
• Decrypt by adding inverse.

4
Modular Multiplication

• Multiplication modulo K
• Multiplicative inverse multiplication mod K
  yields 1
• Only some numbers have inverse
• Use Euclids algorithm to find inverse
• Given x, n, it finds y such that x?y mod n 1
• All number relatively prime to n will have mod n
  multiplicative inverse

5
Totient Function

• x, m relative prime no other common factor than
  1
• Totient function ø(n) number of integers less
  than n relatively prime to n
• if n is prime, ø(n)n-1
• if np?q, and p, q are primes, ø(n)(p-1)(q-1)

6
Modular Exponentiation

• xy mod n xy mod ø(n) mod n
• if y 1 mod ø(n) then xy mod n x mod n

7
RSA (Rivest, Shamir, Adleman)

• The most popular one.
• Support both public key encryption and digital
  signature.
• Assumption/theoretical basis
• Factoring a big number is hard.
• Variable key length (usually 512 bits).
• Variable plaintext block size.
• Plaintext must be smaller than the key.
• Ciphertext block size is the same as the key
  length.

8
What Is RSA?

• To generate key pair
• Pick large primes (gt 256 bits each) p and q
• Let n pq, keep your p and q to yourself!
• For public key, choose e that is relatively
  prime to ø(n) (p-1)(q-1), let pub lte,ngt
• For private key, find d that is the
  multiplicative inverse of e mod ø(n), i.e., ed
  1 mod ø(n), let priv ltd,ngt

9
How Does RSA Work?

• Given pub lte, ngt and priv ltd, ngt
• encryption c me mod n, m lt n
• decryption m cd mod n
• signature s md mod n, m lt n
• verification m se mod n

10
Why Does RSA Work?

• Given pub lte, ngt and priv ltd, ngt
• n pq, ø(n) (p-1)(q-1)
• ed 1 mod ø(n)
• xe?d x mod n
• encryption c me mod n
• decryption m cd mod n me?d mod n m mod n
  m (since m lt n)
• digital signature (similar)

11
Why Is RSA Secure?

• Factoring 512-bit number is very hard!
• But if you can factor big number n then given
  public key lte,ngt, you can find d, hence the
  private key by
• Knowing factors p, q, such that, n pq
• Then ø(n) (p-1)(q-1)
• Then d such that ed 1 mod ø(n)

12
Diffie-Hellman Key Exchange

• Shared key, public communication
• No authentication of partners
• Whats involved?
• P is a prime (about 512 bits), and g lt p
• P and g are publicly known

13
Diffie-Hellman Key Exchange

• Procedure
• Alice Bob
• pick secret Sa randomly pick secret Sb
  randomly
• compute TAgSa mod p compute TBgSb mod p
• send TA to Bob send TB to Alice
• compute TBSa mod p compute TASb mod p
• Alice and Bob reached the same secret gSaSb mod
  p, which is then used as the shared key.

14
DH Security - Discrete Logarithm Is Hard

• T gs mod p
• Conjecture given T, g, p, it is extremely hard
  to compute the value of s (discrete logarithm)

15
Diffie-Hellman Scheme

• Security factors
• Discrete logarithm very difficult.
• Shared key (the secret) itself never transmitted.
• Disadvantages
• Expensive exponential operation
• DoS possible.
• The scheme itself cannot be used to encrypt
  anything it is for secret key establishment.
• No authentication, so you can not sign anything

16
Bucket Brigade Attack...Man In The Middle

• Alice Trudy Bob
• gSa123 gSx 654 gSb 255
• 123 --gt 654 --gt
• lt--654 lt--255
• 654Sa123Sx 255Sx654Sb
• Trudy plays Bob to Alice and Alice to Bob

17
Diffie-Hellman in Phone Book Mode

• DH was subject to active man-in-the-middle attack
  because their public key-component was
  intercepted and substituted
• Phone book mode allows everyone to generate the
  public key-component in advance and publish them
  through other reliable means, e.g. ltTBgt for bob
• All communicating parties agree on their common
  ltg, pgt

18
Encryption With Diffie-Hellman

• Everyone computes and publishes ltp, g, Tgt
• TgS mod p
• Alice communicates with Bob
• Alice
• Picks a random secret Sa
• Computes gbSa mod pb
• Use Kab TbSa mod pb to encrypt message
• Send encrypted message along with gbSa mod pb
• Bob
• (gbSa)Sb mod pb (gbSb)Sa mod pb TbSa mod pb
  Kab
• Use Kab to decrypt