A Cryptography Tutorial

1
A Cryptography Tutorial

• Jim Xu
• College of Computing
• Georgia Tech
• http//www.cc.gatech.edu/jx

2
Why Cryptography?

• Network information needs to be communicated
  through insecure channel.
• Stored information may be accessed without proper
  authorization.
• Cryptography is a systematic way to make that
  harder.

3
Common Security Requirements

• Secrecy(encryption)
• Authenticity(signature/encryption)
• Integrity (signature/encryption)
• Non-repudiation (signature)

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What Cryptography can do?

• Encryption only the authorized party can
  understand the encrypted message.
• Signature allow people to verify the
  authenticity of the message.

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Classical Cryptography

• Shift Cipher (a special case used by Caesar)
• Substitution Cipher
• Affine Cipher
• Vigenere Cipher
• Hill Cipher
• Permutation Cipher

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Cryptoanalysis

• Ciphertext-only attack
• Known plaintext attack
• Chosen plaintext attack
• Adaptive Chosen plaintext attack

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Cryptoanalysis

• Shift Cipher English histogram
• Substitution Cipher histogram again
• Affine Cipher histogram
• Vigenere Cipher more complicated stat
• Hill Cipher Known plaintext attack
• Permutation Cipher histogram semantics

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Frequency of Letter Occurance
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How to achieve perfect secrecy?

• One-pad have a key as long as the plaintext
• For example, shift cipher is perfectly secure if
  the key is random and it is only used to encrypt
  one character!
• Spurious keys S(n) gt K/(P(nR))-1
• Unicity distance that n to make S(n) zero

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Modern Cryptography

• Two broad classes
• 1. Shared-key cryptography
• 2. Public-key cryptography

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Shared-key cryptography

• Rooted in computational complexity
• Sender has M
• Sender sends (M XOR f(x, k), x)
• f is a random function
• Algorithms
• DES, Various fishes, Lucifer, Fiestel, AES
  standards (Rijendel), ...

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DES

• A round can be described as
• Li Ri-1
• The key generation is performed
• An initial permutation PC1 which selects 56 bits
  and divide them in two halves
• In each round
• Select 24 bits from each half using a permutation
  function PC2
• Rotate left each half by one or two position

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Rich theory on pseudorandomness

• Pseudorandom number/bit generator
• Pseudorandom functions (ideal cryptographic hash
  functions)
• Stretch a small completely random string into a
  longer but less random string
• Though less random, indistinguishable to naked
  eyes

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Public Key Cryptography

• Public/private key pair
• Only the owner knows the private key, but
  everyone knows the public key
• If the message is encrypted with the private key,
  then everyone with the public key can recover the
  message, but only the owner can generate the
  encrypted message

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Continued

• If the message is encrypted with the public key,
  only the owner can decrypted it using its private
  key
• The first property can be used for signature and
  the second property can be used for encryption.

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Digital signature

• Sender sends M, TE(hash(M), private)
• The receiver compares E(T, public) and compares
  it with hash(M)
• M is considered genuine if they match

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RSA

• Find two big prime numbers p and q
• Let B pq
• Choose private key C to be a number that is
  coprime with (p-1)(q-1)
• Choose public key D such that CD1 mod
  (p-1)(q-1)

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Continued

• Encrypt M TMC (or MD)
• Decrypt M M TD (or TC)
• Theorem (MC)D M mod B
• Why all the numbers that is coprime with B form
  a group, and the size of that group is (p-1)(q-1)

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Security of RSA

• Hinge upon how hard the factorization is
• If one can break down B into p and q
• then finding C CD 1 mod (p-1)(q-1) is easy
• Factorization is found to be quite hard, at least
  for now.

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Cryptographic Protocols

• System needs are more complicated than what the
  primitives can provide
• Improperly designed, be broken even if none of
  the underlying primitives are broken
• Hard to check whether it is properly designed
  (proof logic/model checking/theorem proving
  methods are involved)

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Key exchange

• Diff-Hellman
• Based on the assumption that knowing prime p and
  pn, finding n will be hard
• Allow two party to share a key
• A senders B pa and remembers a
• B senders A pb and remembers b
• Both sides can generate p(ab)
• Third party can not do that!

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Man in the middle

• C can establish a key with both A and B, by
  posing as B and A respectively
• Solution introduce public key or using return
  address as authentication method

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Public Key Infrastructure

• Need this infrastructure to prevent A from
  claiming that B uses the public key that A
  generates
• Both hierachical and flat infrastructure are
  proposed
• Revocation list a major headache

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Advanced Issues

• Group encryption/signature
• Forward security
• Everlasting security