Key Exchange

• Two parties need to agree on a secret key without revealing it to the public or hacker
• Assume Alice and Bob have secret number s1 and s2 resp.
• Let’s agree on a public number: p
• Step 1:
  • Alice: s1, p
  • Bob: s2, p
  • Hacker p
• Step 2:
  • Alice sends: s1 + p ⇒ Bob
  • Bob sends: s2 + p ⇒ Alice
  • Hacker: p, s1 + p, s2 + p
• Step 3:
  • Alice generates: s1 + <Bob message> = s1 + (s2 + p)
  • Bob generates: s2 + <Alice message> = s2 + (s1 + p)
  • Final Secret Key: s1 + s2 + p
• Security issue:
  • Hacker can access In-transit values:
    • p
    • Alice ------s1 + p⇒ Bob
    • Alice ⇐—s2 + p---- Bob
  • Hacker can then find the original value by reversing the addition operation
    • Alice original key = <Alice message> - p = (s1 + p) - p = s1
    • Bob original key = <Bob message> - p = (s2 + p) - p = s2

Diffie Hellman Key Exchange

• aka DHKE or DH
• We want the reversing operation to be harder, this uses modulo operations
• Assume Alice and Bob have secret number s1 and s2 resp.
• Let’s agree on public numbers: b (base), p (modulo)
• Alice and Bob sends

$$\begin{array}{r}\text{Alice to Bob: }{b}^{s1}\mathrm{mod}p\\ \text{Bob to Alice: }{b}^{s2}\mathrm{mod}p\end{array}$$

• Alice and Bob generates:

$$\begin{array}{rl}\text{Alice generates }& =(\text{Bob’s message}{)}^{s1}\mathrm{mod}p\\ & =(b^{s2}{)}^{s1}\mathrm{mod}p\\ \text{Bob generates }& =(\text{Alice’s message}{)}^{s2}\mathrm{mod}p\\ & =(b^{s1}{)}^{s2}\mathrm{mod}p\\ & \\ \text{Final Secret Key}& =b^{(s1\cdot s2)}\mathrm{mod}p\end{array}$$

• Hacker has:
  • b, p, b^s1 mod p, b^s2 mod p
  • It is difficult for hacker to reverse the operation to derive s1 or s2
• Notice:
  • May require lot of modulo calculations
  • p is generally chosen to be prime to make it stronger
  • Achieves forward secrecy, hence used to generate secret key in a session
  • Advanced version is ECDH (Elliptic Curve DH)
• Disadvantages:
  • Vulnerable for Man in the middle Attack (MITM) where the attacker can impersonate Bob/Alice to both sides and generate keys
  • Hence it is needed to exactly know whether the sender really was Alice or Bob not
• References
  • https://www.youtube.com/watch?v=85oMrKd8afY