COMP4690 Tutorial

1
COMP4690 Tutorial

• Cryptography
• Number Theory

2
Outline

• DES Example
• Number Theory
• RSA Example
• Diffie-Hellman Example

3
DES

• Some remarks
• DES works on bits
• DES works by encrypting groups of 64 bits, which
  is the same as 16 hexadecimal numbers
• DES uses keys which are also apparently 64 bits
  long. However, every 8th key bit is ignored in
  the DES algorithm, so the effective key size is
  56 bits.
• If the length of the message to be encrypted is
  not a multiple of 64 bits, it must be padded.
  E.g.
• The plaintext message "Your lips are smoother
  than vaseline" is, in hexadecimal,
  "596F7572206C6970 732061726520736D
  6F6F746865722074 68616E2076617365 6C696E650D0A".
• We then pad this message with some 0s on the end,
  to get a total of 80 hexadecimal digits
  "596F7572206C6970 732061726520736D
  6F6F746865722074 68616E2076617365
  6C696E650D0A0000".
• Then apply DES.

4
Key generation example

• Let K be the hexadecimal key K
  133457799BBCDFF1. This gives us as the binary key
  
• K 0001 0011 0011 0100 0101 0111 0111 1001 1001
  1011 1011 1100 1101 1111 1111 0001
• 16 subkeys (48-bit) will be generated from K.

5
Key generation example

• Based on table PC-1 (Permuted Choice 1), we get
  the 56-bit permutation
• K 1111000 0110011 0010101 0101111 0101010
  1011001 1001111 0001111
• Next, split this key into left and right halves,
  C0 and D0, where each half has 28 bits.
• C0 1111000 0110011 0010101 0101111 D0
  0101010 1011001 1001111 0001111

6
Key generation example

• we now create sixteen blocks Cn and Dn, 1ltnlt16.
  Each pair of blocks Cn and Dn is formed from the
  previous pair Cn-1 and Dn-1, respectively, for n
  1, 2, ..., 16, using a schedule of left
  shifts".

Round number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Bits rotated 1 1 2 2 2 2 2 2 1 2 2 2 2 2 2 1
7
Key generation example

• C0 1111000011001100101010101111D0
  0101010101100110011110001111
• C1 1110000110011001010101011111D1
  1010101011001100111100011110
• C2 1100001100110010101010111111D2
  0101010110011001111000111101
• C3 0000110011001010101011111111D3
  0101011001100111100011110101

8
Key generation example

• We now form the subkeys Kn, for 1ltnlt16, by
  applying the table PC-2 (Permutation Choice Two)
  to each of the concatenated pairs CnDn.
• For the first subkey, we have
• C1D1 1110000 1100110 0101010 1011111 1010101
  0110011 0011110 0011110
• After we apply the permutation PC-2
• K1 000110 110000 001011 101111 111111 000111
  000001 110010

9
Modular Arithmetic

• Two integers a and b are said to be congruent
  modulo n, if
• (a mod n) (b mod n)
• This is written as ab mod n
• Define Zn as the set of nonnegative integers less
  than n Zn0,1,,(n-1)

10
Modular Arithmetic

• Properties of modular arithmetic

11
Modular Arithmetic

• Define Zp as the set of nonnegative integers less
  than a given prime number p Zp0,1,,(p-1)
• Because p is prime, all of the nonzero integers
  in Zp are relatively prime to p.
• There exists a multiplicative inverse for all of
  the nonzero integers in Zp
• For each nonzero w in Zp, there exists a z in Zp
  such that w x z 1 mod p. z is called the
  multiplicative inverse of w. Or, z w-1.

12
Number Theory

• Fermats Little Theorem
• ap-1 1 mod p
• where p is prime and gcd(a,p)1
• E.g.
• a 7, p 19
• 724911 mod 19
• 741217 mod 19
• 784911 mod 19
• 7161217 mod 19
• ap-1718716x727x11771 mod 19

13
Number Theory

• An alternative form of Fermats Little Theorem
• ap a mod p
• where p is prime and a is any positive integer
• E.g.
• p5,a3,352433 mod 5
• p5,a10,10510000010 mod 50

14
Number Theory

• Eulers Totient Function ø(n)
• The number of positive integers less than n and
  relatively prime to n
• For prime number p,
• ø(n) p 1
• For n pq where p and q are two different prime
  numbers
• ø(n) (p 1) (q 1)

15
Number Theory

• Example ø(21)
• From 1 to 21, totally 21 numbers
• 21 3x7, 3 and 7 are prime
• 3s multiples
• 3, 6, 9, 12, 15, 18, 21
• 7s multiples
• 7, 14, 21
• Other numbers are all relatively prime to 21
• 21-7-31 (3-1)x(7-1)

16
Number Theory

• Eulers Theorem
• aø(n) 1 mod n
• where gcd(a,n)1
• E.g.
• a3n10 ø(10)4
• hence 34 81 1 mod 10
• a2n11 ø(11)10
• hence 210 1024 1 mod 11

17
Number Theory

• The powers of an integer a, modulo n
• a, a2, a3, (mod n)
• If a and n are relatively prime, based on Eulers
  theorem, we have aø(n) 1 mod n
• a, a2, a3, will have a repeated pattern
• E.g., ø(5)4, 3ø(5)811 mod 5
• 3, 4, 2, 1, 3, 4, 2, 1,
• There may exist lots of m such that am 1 mod n
• The least positive exponent m such that am 1
  mod n is referred to as
• the order of a (mod n)
• the exponent to which a belongs (mod n)
• the length of the period generated by a

18
Number Theory
19
Number Theory

• Primitive root
• If a numbers order (mod n) is ø(n), this number
  is called a primitive root of n
• Property of primitive root
• If a is a primitive root of n, then its powers
  a,a2, a3,, aø(n) are distinct (mod n), and are
  all relatively prime to n.
• In particular, for a prime number p, if a is a
  primitive root of p, then a,a2, a3,, ap-1 are
  distinct (mod p).
• From the previous table, we can see that prime
  number 19s primitive roots are 2, 3, 10, 13, 14,
  and 15.

20
RSA Example

• Select primes p17 q11
• Compute n pq 1711187
• Compute ø(n)(p1)(q-1)1610160
• Select e gcd(e,160)1 choose e7
• Determine d de1 mod 160 and d lt 160
• d23 since 237161 101601
• Publish public key KU7,187
• Keep secret private key KR23,17,11

21
RSA Example

• given message M 88
• encryption
• C 887 mod 187 11
• decryption
• M 1123 mod 187 88

22
RSA Example

• Fast Modular Exponentiation
• To calculate 887 mod 187
• 881 mod 187 88
• 882 mod 187 7744 mod 187 77
• 884 mod 187 772 mod 187 132
• 887 mod 187 88421 mod 187 132x77x88 mod 187
  894,432 mod 187 11
• To calculate 1123 mod 187
• 111 mod 187 11
• 112 mod 187 121
• 114 mod 187 14,641 mod 187 55
• 118 mod 187 552 mod 187 33
• 1116 mod 187 332 mod 187 154
• 1123 mod 187 1116421 mod187 154x55x121x11
  mod 187 11,273,570 mod 187 88

23
Diffie-Hellman Key Exchange
24
Diffie-Hellman Key Exchange

• users Alice Bob who wish to swap keys
• agree on prime q7 and a5
• select random secret keys
• A chooses xA3, B chooses xB2
• compute public keys
• yA53 mod 7 6 (Alice)
• yB52 mod 7 4 (Bob)
• compute shared session key as
• Alice KAB yBxA mod 7 43 mod 7 1
• Bob KAB yAxB mod 7 62 mod 7 1

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Diffie-Hellman Key Exchange

• users Alice Bob who wish to swap keys
• agree on prime q353 and a3
• select random secret keys
• A chooses xA97, B chooses xB233
• compute public keys
• yA397 mod 353 40 (Alice)
• yB3233 mod 353 248 (Bob)
• compute shared session key as
• Alice KAB yBxA mod 353 24897 mod 353 160
• Bob KAB yAxB mod 353 40233 mod 353 160