Public Key Cryptography

• The system was proposed by Diffe and Hellman in 1976.
  • Involves a public key $k_{PUB}$ and an private key $k_{PRIV}$
  • $P = D(k_{PUB} , E(k_{PRIV}, P)) $
  • $P = D(k_{PRIV} , E(k_{PUB}, P)) $
• RSA encryption was the first public implementation in 1977
• basic observation
  • Factoring is hard.
  • $ (m^e)^d \equiv m % n$
  • It is hard to find e given n and d.
  • Also $(m^e)^d = (m^d)^e$
  • So e and d can be keys.
• RSA Keys
  • Need to be at least 256 bit long
  • But are frequently up to 2000 bits long.
• This makes it a computationally expensive computation
  • IE It takes a long time.
  • It is probably not viable for regular message exchange.
  • It takes as much as 10,000 times longer to encrypt/decode than DES/AES
• Communications using Public/Private key systems.
  • Assume a pair of people who wish to communicate (Alice and Bob)
  • An open communications network
  • And an malicious actor Eve.
• A problem: How do Alice and Bob exchange messages?
  • Bob creates a public key $k_{PUB-B}$ and an private key $k_{PRIV-B}$
  • Alice creates a public key $k_{PUB-A}$ and an private key $k_{PRIV-A}$
  • Each publishes their pubilc key
  • Each hides their private key.
  • Bob picks $k_{SYM}$ for the exchange of information.
• Solution 1:
  • Bob creates $M=E(k_{PRIV-B},k_{SYM})$
  • Bob sends M to Alice
  • Alice decodes M, $k_{SYM} = E(k_{PUB-B},M)$
  • Alice and bob happily exchange messages using AES and $k_{SYM}$.
• Problem 1:
  • Unfortunately, there is nothing to stop Eve from decoding M as well.
  • She knows Bob's public key.
• The book walks through a scenario where Eve actually intercepts the exchange of public keys.
  1. Bob requests Alice's public key.
  2. Eve intercepts this message and sends a message as Bob requesting Alice's public key
  3. Eve receives Alice's public key but sends a new public key to Bob.
  4. Even Intercepts the message containing $k_{SYM}$ above, decodes, and encodes it and re transmits it to Alice.
  5. Bob and Alice exchange messages and Eve listens.
• A much better scenario is where both Alice and Bob publish their public keys.
  • The keys must be public and conformable.
  • IE no transmitting fake public keys.
  1. Bob produces $M=E(k_{PUB-A}, E(k_{PRIV-B}, k_{SYM}))$
  2. Bob sends M to Alice.
  3. Alice can decode M, $k_{SYM} =E(k_{PUB-B},(k_E{PRIV-A},M ))$
     • Alice knows this message came from Bob, because only Bob's private key can be decoded with Bob's public key.
     • No one but Alice can read the message because it was decoded only with her private key.
• They introduce the idea of a nonce
  • This is just a nonsense message.
  • But it can be used to verify authenticity of both sides.
  • Using the system above Bob and Alice exchange the nonce - twice.
  • This will assure symmetric authentication.
• But how do we assure that the public keys are authentic?
  • A Certification Authority (CA) is a trusted third party.
  • They profice digital certification of the public key.
  • Thus ensure trust of the private key.
• In one case
  • Bob registeres his public key with a CA
  • The CA issues Bob a certificate,
    • This could be Bob's public key encrypted with the CA's private key.
  • To interact with Bob, Alice asks Bob for his Public Key and his certificate.
  • She decrypts the certificate with the CA's public key
  • And compares this to Bob's public key
  • Or just needs the certificate, from which she can extract Bob's public key.
• There are other scenarios, but in general you need a CA you can trust.