Introduction to Cryptography

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Introduction to Cryptography

• By Yan Zhou

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Outline

• Encryption and Decryption
• Cryptosystems
• Symmetric
• Public-Key
• Hybrid
• Summary

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Cryptography
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Encryption

Alice wewillmeetatmidnight
Ek1(M) C
hphtwwxppelextoytrse
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Decryption
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Algorithms and Keys

• Restricted Algorithm
• E(M) C
• D(C) M
• Key
• Symmetric Algorithm (Single Key)
• Ek(M) C
• Dk(C) M
• Public-Key Algorithm (Two key)
• Ek1(M) C
• Dk2(C) M

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Shift Cipher

• For 0 ? K ? 25, define
• Ek(x) x K mod 26
• And
• Dk(y) y - K mod 26

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An example of shift cipher

• Alice K 11
• wewillmeetatmidnight

hphtwwxppelextoytrse
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Protocol Using Symmetric Cryptography

• Alice and Bob agree on a cryptosystem
• Alice and Bob agree on a key
• Alice encrypts and sends her message to Bob
• Bob decrypts the message and reads it

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Problems with Symmetric Cryptosystems

• Keys must be distributed in secret
• If a key is compromised, messages can be
  decrypted and produced
• The total number of keys n(n-1)/2 increases
  rapidly as the number of users n increases.

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RSA

• Asymmetric algorithm, the encryption (public) key
  and the decryption (private) key are different.
• Security is based on the difficulty of factoring
  large numbers

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RSA Encryption

• Public key
• n product of two random large primes, p and q
• (p and q must remain secret)
• e randomly chosen, relatively prime to (p-1)(q-1)
• Private key
• d e-1 mod ((p-1)(q-1))
• Encrypting C Me mod n
• Decrypting M Cd mod n

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An example of RSA

• Suppose Bob picks p 101 and q 113, then
• n pq 11413
• And
• (p-1)(q-1) 11200 26527
• e can be used as a public key if and only if it
  is not divisible by 2, 5 and 7.
• Suppose Bob chooses e 3533. Then
• d 3533-1 mod 11200 6597
• Bob publishes e and n, and keeps d secret.

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Example Continues

• Suppose Alice wants to send 9726 to Bob. She will
  compute and send
• 97263533 mod 11413 5761
• Bob will decrypt the message
• 57616597 mod 11413 9726

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Protocol Using Public-Key Cryptography

• Alice and Bob agree on a public-key cryptosystem
• Bob sends Alice his public key
• Alice encrypts her message using Bobs public key
  and sends it to Bob
• Bob decrypts Alices message using his private key

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Problems with Public-Key Cryptosystem

• Slow
• Key management

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Hybrid Cryptosystem

• Bob sends Alice his public key
• Alice generates a random session key, K, encrypts
  it using Bobs public key, and sends it to Bob
• Bob decrypts Alices message using his private
  key to get the session key
• Alice and Bob encrypt their messages using the
  session key

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Man-in-the-Middle Attack

• Bob sends Alice his public key. Eve intercepts
  this key and sends Alice her own public key
• Alice generates a random session key, K, encrypts
  it using Bobs public key, and sends it to Bob.
  Eve intercepts and decrypts the message to get
  the session key K, re-encrypts the message with
  Bobs public key and sends it to Bob.
• Bob sends messages to Alice encrypted with K. Eve
  can read messages between Bob and Alice

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Interlock Protocol

• Alice sends Bob her Public Key
• Bob sends Alice his Public key
• Alice encrypts her message using Bobs public key
  and sends half of the encrypted message to Bob
• Bob encrypts his message using Alices public key
  and sends half of the encrypted message to Alice
• Alice sends the other half of her encrypted
  message to Bob
• Bob puts the two halves of Alices message
  together and decrypts it with his private key.
  Bob sends the other half of his encrypted message
  to Alice
• Alice puts the two halves of Bobs message
  together and decrypts it with her private key.

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Summary

• Encryption and Decryption
• Cryptosystems
• Algorithms
• Shift cipher
• RSA
• protocols
• Symmetric
• Public-Key
• Hybrid

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Book References

• Applied Cryptography, Protocols, Algorithms, and
  Source Code in C (Second Edition), by Bruce
  Schneier. John Wiley and Sons, 1995.
• Cryptography, Theory and Practice, by Douglas, R.
  Stinson. CRC Press, 1995.
• Handbook of Applied Cryptography, by Alfred J.
  Menezes, Paul C. van Oorschot and Scott A.
  Vanstone. CRC Press, 1996.