Computer Security Lecture 22

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Computer Security Lecture 22

• Phillip G. Bradford
• Computer Science
• University of Alabama

2
Impact of Background and Experience on Software
Inspections
Dr. Jeffrey Carver Department of Computer
Science University of Maryland

• 11 AM, Friday March 7, 2003
• Houser 108

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Outline

• Review Last Lecture
• Overview of the next lectures
• RC5
• Public Key Systems
• Diffie-Hellman
• RSA

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Objectives

• Understand the RC5 Hash Function
• Public Key systems
• Background for
• Diffie-Hellman
• Diffie-Hellman
• Background for
• RSA
• RSA in next lecture

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Rivests RC5See also RSA Research Web

• Hash function
• Xor, addition and rotation
• Variable number of rounds r
• Uses 2(r 1) key rounds
• S1,,S2(r1)
• Plaintext represented as LR

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Rivests RC5

• First L L S0 and R R S1
• For i ? 1 to r do
• L ? ((L xor R) ltltlt R) S2i
• R ? ((R xor L) ltltlt L) S2i1
• Where ltltlt is a right circular shift

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Rivests RC5

• Generating the keys, given T holding initial key
• P and Q are constants
• See page 345
• S0 P
• For i ? 1 to 2(r1) do
• Si ? (Si-1 Q) mod 232

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Rivests RC5

• Mixing Key T, T is 32c for some const c
• i ? j ? 0 A ? B ? 0
• Loop 3n times (n max2(r1), c )
• A ? Si ? (Si A B) ltltlt 3
• B ? Ti ? (Ti A B) ltltlt (A B)
• i ? (i1) mod 2(r1)
• j ? (j1) mod c

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Diffie-Hellman Background

• Primitive Roots
• Let p be prime, then the integer g lt p
• Is a primitive root iff
• For all a in Zp
• Then there is an j such that gj a mod p
• In other words, g is a generator of Zp

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Diffie-Hellman Background

• Examples, (page 253) where p 11
• 21 2 mod 11
• 22 4 mod 11
• 23 8 mod 11
• 24 5 mod 11
• 25 10 mod 11
• 26 9 mod 11
• 27 7 mod 11
• 28 3 mod 11
• 29 6 mod 11
• 210 1 mod 11

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Diffie-Hellman Background

• Not all elements in Zp are generators!
• Example 3 is not primitive mod 11
• Since 3j 2 mod 11 does not hold for any j
• General algorithm for primitive roots
• Take p-1s factors, say p-1 q1qn
• Let d (p-1)/q, for each q in q1 , , qn
• Then if gd 1 mod p, then g is not a generator

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Diffie-Hellman Background

• How can we find a primitive root?
• Famous Theorem
• If p gt 2 is prime, then it has Phi(p-1) primitive
  roots mod p.
• Phi(n) is Eulers Phi-function
• Number of integers between 1 and n that are
  relatively prime to n
• Give a primitive root hunting algorithm!

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Diffie-Hellman

• Alice Bob agree on p and g that is a primitive
  root mod p
• Allow p and g to be public, perhaps with
  verifying proofs

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Diffie-Hellman

• Alice chooses a large random integer x
• Sends Bob X gx mod p x secret
• Bob chooses a large random integer y
• Sends Alice Y gy mod p y secret
• Alice now computes k Yx mod p
• Bob now computes k Xy mod p
• Note secret key k k gxy mod p

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Review of Public Keys

• Two keys
• The public Key is open to the world
• The private key is secret, only you know it
• Knowing the public key, it is intractable to
  compute the private key
• Each key is the others inverse
• In terms of the cipher

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Background Review for RSA

• Eulers Phi Function Again!
• Phi(n) The number of relatively prime integers
  to n
• If p is prime, then Phi(p) p-1
• If p and q are prime
• Then Phi(pq) (p-1)(q-1)
• If p and q are relatively prime
• Then Phi(pq) Phi(p) Phi(q)

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RSA Algorithm

• Find to (large) primes p and q, let npq
• Compute Phi(n) (p-1)(q-1)
• Find an integer exponent E so that
• gcd(Phi(n),E) 1 relative primality
• Compute d E-1 mod Phi(n)
• Public Key KU ltE,ngt
• Private Key KR ltd,ngt

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RSA Encryption Decryption

• Encryption
• Given a block of plaintext M lt n
• Ciphertext C ME mod n
• Decryption
• M Cd mod n
• Next lecture Why this works!