R S A

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R S A

• POON TENG HIN

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Main topic

• RSA
• Shamirs Three-Pass Protocol
• Other issues

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A IQ question
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Encryption

• The locks in computer network
• 1-1 mapping function f
• so that c f(m)

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Decryption

• The keys in computer network
• so that 
• f-1(c) f-1(f(m)) (f-1f)(m) m

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RSA We need

• Function
• Modulo Operation
• Greatest Common Divisor
• Multiplicative Inverse
• Number theory 
• Prime number

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Multiplicative Inverse

• (x y) mod n 1.
• The integer y is called a multiplicative inverse
  of x, usually denoted x-1 (it
• is unique if it exists).

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Prime number

• People keep finding large prime numbers for
  computer Security.
• How the prime number are
• used?

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RSA

• RSA is an algorithm for public-key cryptography
• By Ron Rivest, Adi Shamir, Leonard Adleman

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Many application

• Because of security, high strength
• Encryption
• Digital signatures
• E.g electronic transactions,
• software certification.

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RSA encryption and decryption

• Encryption C Me mod n
• Decryption M Cd mod n

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Lets try it

• ABCDEFGHIJKLMNOPQRSTUVWXYZ
• 123426
• Public key n 35, e 5 C Me mod n
• Private key d 5 M Cd mod n
• My word
• 17 21 14 33 8
• 6 30 11
• Also, try to give me your words

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The Security of the RSA

• p, q, (n) must be kept secret.
• It is believed that determine (n) given n is
  equivalent to factoring n.
• With presently known algorithms, determining d
  given e and n, appears to be at least as
  time-consuming as the factoring problem.
• So use factoring as the benchmark for security
  evaluation.

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ASCII

• http//www.cs.drexel.edu/jpopyack/IntroCS/HW/ASCI
  I.html
• A website of ASCII code

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Term

• Plaintext M ( M 0,1)
• Cipher text C (C 0,1)
• It needs two distinct primes p and q
• F(n) (p-1)(q-1)
• select an integer e such that gcd(e, F(n)) 1
• Where n pq, ngtM
• Compute the d where ed 1 (mod F(n))
• Public key (e,n)
• Private key d

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n p and q

• Randomly choose p and q
• And n p X q
• A sample n from
• http//www.rsa.com/rsalabs/node.asp?id2093
• RSA-576
• 1881988129206079638386972394616504398071635633794
  17382700763356422988859715234665485319060606504743
  04531738801130339671619969232120573403187955065699
  6221305168759307650257059

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e

• gcd(e, F(n)) 1 and e gt 1
• A table to find e and d

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Eulers Totient Functopm F(n)

• F(n) is the number of positive integers less than
  n that is relative prime to n
• Example F(6)
• the GCD(x,6) 1 when x 1,5
• so F(6) 2

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Eulers Totient Function F(n)

• F(p) p-1 for any prime number p
• F(pq) (p-1)(q-1) for any two distinct primes p
  and q

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Fermats and Eulers Theorem

• Eulers For every integer a and n that are
  relatively prime,
• aF(n)mod n 1
• Fermats
• If n p is prime,
• ap-1 mod p 1

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d

• ed 1 (mod F(n)) or d e-1 mod n
• Such that ex F(n) y 1 and d is the value of x
• One of the method is Euclidean algorithm
• http//www.di-mgt.com.au/euclidean.html

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d example

• Fo example F(n) 20, e 3
• Firstly, gcd(20,3) 1 if the inverse exists.
• We use Euclidean algorithm
• 20 3 x 6 2
• 3 2 x 1 1
• 1 3 1X2
• 3 1 X (20 6 X 3)
• -1 X 20 7 X 3 (ex ny 1)
• so d 7

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Another example ofEuclidean algorithm

• 66 1 35 31 gcd(35, 31)
• 35 1 31 4 gcd(31, 4)
• 31 7 4 3 gcd(4, 3)
• 4 1 3 1 gcd(3, 1)
• 3 3 1 0 gcd(1, 0)
• So,
• gcd(66, 35) gcd(35, 31) gcd(31, 4) gcd(4,
  3) gcd(3, 1) gcd(1, 0) 1.

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See it again

• Encryption C Me mod n
• Decryption M Cd mod n
• Needs two distinct primes p and q
• And F(n) (p-1)(q-1)
• select an integer e such that gcd(e, F(n)) 1
• Where n pq, ngtM
• Compute the d where ed 1 (mod F(n))
• Public key (e,n)
• Private key d

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RSA calculation

• http//www-cs-students.stanford.edu/tjw/jsbn/rsa2
  .html
• http//www.cs.drexel.edu/jpopyack/IntroCS/HW/RSAW
  orksheet.html

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Correctness of Decryption
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Correctness of Decryption
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Correctness of Decryption
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Answer of IQ question

• 1.A lock the box by his lock A
• 2.A-------------? B (Box with lock A)
• 3.B lock the box by his lock B
• 4.B---------------?A (Box with lock A B)
• 5.A unlock his lock A
• 6.A ---------------? B (Box with lock B)
• 7. B unlock his lock B
• finish

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Shamirs Three-Pass Protocol

• This is the protocol similar to the answer of the
  IQ question
• This is different to RSA
• In this protocol, we need a prime p which is a
  public knowledge.

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A and B

• A selects a random number a with
• gcd(a, p-1) 1
• B selects a random number b with
• gcd(b,p-1) 1
• a-1 and b-1
• are the inverse of a and b of mod p-1

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The protocol

• A computes k1 ka mod p and send k1 to B
• B computes k2 k1b mod p and send k2 to A
• A computes k3 k2a-1mod p and send k3 to B
• Finally, B computes k k3b-1 mod p
• and get k.

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Homework

• Q1.Using slide 13, what is the message under
• 12 21 10 24 20 4 15 14
• 15 14 10
• 4 24
• 6 4 14 4 24 8 10 9
• Q2. Find d if F(n) 58, e 27
• (use Euclidean algorithm)

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Others

• Others issues I would like to share.
• I suggest you may think about them.

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Comp364

• Computer and Communications Security COMP364
• By Prof. Cunsheng Ding

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Bridge

• People like math will like this game.

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Classical One-key Cipher or Cryptosystem

• Encryption c Ek(m), where Ek is usually
  applied to blocks of the plaintext m.
• Decryption m Dk(c), where Dk is usually
  applied to blocks or characters of the ciphertext
  c.

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Example Transposition Ciphers

• Example Let d 4 and define f by
• i 0 1 2 3
• f(i) 2 0 3 1
• Then f is a permutation of Z4.
• The inverse permutation f-1 is given by
• i 0 1 2 3
• f-1(i) 1 3 0 2

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Simple Substitution Ciphers

• E.g

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http//www.blog.republicofmath.com/archives/4120
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Is that a Paradox?
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Example (Condorcet, 17851994)

• A B C
• 1 plan1 plan3 plan2
• 2 plan2 plan1 plan3
• 3 plan3 plan2 plan1
• Conclusion
• Most people think that
• plan1 is better than plan2
• plan2 is better than plan3
• plan3 is better than plan1

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END

• ByeBye