Chapter 3 Encryption Algorithms

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Chapter 3Encryption Algorithms Systems (Part
C)
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Outline

• NP-completeness Encryption
• Symmetric (secret key) vs Asymmetric (public key)
  Encryptions
• Popular Encryption Algorithms
• Merkle-Hellman Knapsacks
• RSA Encryption
• El Gamal Algorithms
• DES
• Hashing Algorithms
• Key Escrow Clipper

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RSA Encryption

• 1978 Rivest, Shamir, Adelman
• Public key encryption
• Remains secure to date
• Encryption key (e) and decryption key (d) are
  interchangeable.
• The two keys, e and d, are carefully chosen such
  that C Pe mod n (encryption) and P Cd mod n
  (decryption).

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Euler Totient Function

• ?(n) the number of positive integers less than n
  and are relatively prime to n.
• If n is prime
• ?(n) n 1
• When n p q, where both p and q are primes and
  p ? q
• ?(n) ?(p) ?(q) (p 1) (q 1)

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RSA Encryption

• Public key (e, n)
• Private key (d, n)
• Step 1 Choose n, p, q
• n p q, where both p and q are primes and p ?
  q
• Example n 143 p q 11 13

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RSA Encryption

• Step 2 Choose e.
• e is relatively prime to ?(n). That is, e is
  relatively prime to (p-1)(q-1).
• Example e 17, which is relatively prime to
  1012.
• Step 3 Compute d.
• d is the inverse of e mod (p-1)(q-1).
• Use the algorithm on page 81 to compute inverses.
• Note A Java implementation of the algorithm is
  available at the class page.
• Example d e-1 mod (p-1)(q-1) 17-1 mod 120
  113

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RSA Encryption

• An example (pp.94-95) P 7
• Let n 143, p 11, q 13, and e 11.
• Note e is relprime to (p-1)(q-1).
• Then d 11
• Note d is the inverse of e mod (p-1)(q-1).
• Encryption
• C Pe mod n 711 mod 143 106
• Decryption
• P Cd mod n 5011 mod 143 7

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RSA Encryption

• Another example P 7
• Let n 143, p 11, q 13, and e 17.
• Note e is relprime to (p-1)(q-1).
• Then d 113
• Note d is the inverse of e mod (p-1)(q-1).
• Encryption
• C Pe mod n 717 mod 143 50
• Decryption
• P Cd mod n 50113 mod 143 7

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RSA Encryption

• Still another example P 55
• Let n 285, p 19, q 17, and e 37.
• Note e is relprime to (p-1)(q-1), 288.
• d 109
• Note d is the inverse of e mod (p-1)(q-1).
• Encryption
• C Pe mod n 5537 mod 288 55
• Decryption
• P Cd mod n 55109 mod 288 55

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RSA Encryption

• The cryptographers job
• Find three primes, p, q, and e, where
• p q n and
• e is relatively prime to (p-1)(q-1).
• Compute d based on e and n.
• The challenge p, q, and e must be large enough
  primes.
• See discussions on p.95.

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RSA Encryption

• The cryptanalysts job
• P Cd mod n
• Available (e, n).
• Find two primes p and q, such that p q n and
  e is relatively prime to (p-1)(q-1).
• Compute d d inverse (e, (p-1)(q-1))
• Q Wheres the secrecy?
• Q Given n and a prime e, how hard is it to find
  two distinct primes, p and q, such that pq n
  and (p-1)(q-1) is relprime to e?

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El Gamal Algorithm

• A public key algorithm
• 1984
• Important in the U.S. DSS (Digital Signature
  Standard)
• Digital Signatures
• The sender computes the digital signature using
  his own private key.
• DS E (Keypriv, P)
• The receiver verifies the signature using the
  senders public key.
• P D (Keypub, DS)

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El Gamal Algorithm

• To generate a key pair
• Choose a prime p and two integers, a and x, such
  that a lt p and x lt p.
• The prime p should be chosen so that (p-1) has a
  large prime factor q.
• Calculate the public key y ax mod p.
• Private key x
• Public key y

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El Gamal Algorithm

• (The sender) To sign a message m
• Choose a new random integer k, 0 lt k lt p-1 and k
  is relprime to (p-1).
• Compute r ak mod p.
• Compute s k-1 ( m xr ) mod (p-1)
• The message signature r and s.
• Verification A recipient use the public key (y)
  to compute ( y r r s ) mod p and determine if it
  is equivalent to am mod p.

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Hash Algorithm

• A hash algorithm is a check function that
  protects data against modifications.
• C.f., checksum in network transmission
• Hash functions produce a reduced form of a body
  of data (called a digest or check value) such
  that most changes to the data will also change
  the reduced form.
• A cryptographic hash function uses a
  cryptographic function as part of the hash
  function.
• 1992 Secure Hash Algorithm (SHA)

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Secure Hash Algorithm (SHA)

• 1992 NIST
• Input data lt 264 bits
• 160-bit digest
• Strength diffusion, the avalanche effect
• See Fig. 3-9, p.99
• C.f., MD4, MD5
• Both MD5 and SHA are variants of the MD4 by
  Rivest.
• Strength MD4 lt MD5 lt SHA

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Summary

• Public key encryption algorithms Merkle-Hellman,
  RSA, El Gamal
• SHA
• Next DES, Key Escrow