1. Prime and Relatively Prime Numbers
Divisors: We say that b  0 divides a if a = mb for some m, where a, b and m are integers.
b divides a if there is no remainder on division.
The notation b|a is commonly used to mean that b divides a.
If b|a, we say that b is a divisor of a.
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2. Prime and Relatively Prime Numbers (cont’d)
If a|1, then a =  1.
If a|b and b|a, then a =  b.
Any b  0 divides 0.
If b|g and b|h, then b|(mg + nh) for arbitrary integers m and n.
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3. Prime and Relatively Prime Numbers (cont’d)
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4. Prime and Relatively Prime Numbers (cont’d)
Table 7.1 Primes under 2000
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5. Prime and Relatively Prime Numbers (cont’d)
The above statement is referred to as the prime number theorem, which was proven in 1896 by Hadaward and Poussin.
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6. Prime and Relatively Prime Numbers (cont’d)
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7. Prime and Relatively Prime Numbers (cont’d)
Whether there exists a simple formula to generate prime numbers?
An ancient Chinese mathematician conjectured that if n divides 2n - 2 then n is prime. For n = 3, 3 divides 6 and n is prime. However, For n = 341 = 11  31, n dives
Mersenne suggested that if p is prime then Mp = 2p - 1 is prime. This type of primes are referred to as Mersenne primes. Unfortunately, for p = 11, M11 = = 2047 = 23  89.
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8. Prime and Relatively Prime Numbers (cont’d)
Fermat conjectured that if Fn = 22n + 1, where n is a non-negative integer, then Fn is prime. When n is less than or equal to 4, F0 = 3, F1 = 5, F2 = 17, F3 = 257 and F4 = are all primes. However, F5 = = 641  is not a prime bumber.
n2 - 79n is valid only for n < 80.
There are an infinite number of primes of the form 4n + 1 or 4n + 3.
There is no simple way so far to gererate prime numbers.
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9. Prime and Relatively Prime Numbers (cont’d)
Factorization of an integer as a product of prime numbers
Example: 91 = 7  13; = 7  112  13.
Useful for checking divisibility and relative primality to be discussed later.
Factorization is in gereral difficult.
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10. Prime and Relatively Prime Numbers (cont’d)
Define notation gcd(a,b) to mean the greatest common divisor of a and b.
The positive integer c is said to be the gcd of a and b if
c|a and c|b
any divisor of a and b is a dividor of c.
Equivalently, gcd(a,b) = max[k, such that k|a and k|b]
gcd(a,b) = gcd(-a,b) = gcd(a,-b) = gcd(-a,-b) =gcd(|a|,|b|)
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11. Prime and Relatively Prime Numbers (cont’d)
gcd(a,0) = |a|.
Factorization is one possible but in general inefficient way to calculate gcd. Whereas, Euclid‘s algorithm (to be discussed later) is more efficient.
Relative primality
the integers a and b are relatively prime if they have no prime factors in common
or equivalently, their only common factor is 1
or equivalently, gcd(a,b) = 1
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12. Information Security -- Public-Key Cryptography
Modular Arithmetic
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13. Modular Arithmetic (cont’d)
Examples:
a = 11; n = 7; 11 = 1  7 + 4; r = 4.
a = -11; n = 7; -11 = (-2)  7 + 3; r = 3.
If a is an integer and n is a positive integer, define a mod n to be the remainder when a is divided by n.
Then, a = a/n  n + (a mod n);
Example: 11 mod 7 = 4; -11 mod 7 = 3.
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14. Modular Arithmetic (cont’d)
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15. Modular Arithmetic (cont’d)
Properties of modular arithmetic operations
Proof of Property 1:
Define (a mod n) = ra and (b mod n) = rb. Then a = ra + jn and b = rb + kn for some integers j and k. Then,
(a+b) mod n = (ra + jn + rb + kn) mod n
= (ra + rb + (j + k)n) mod n
= (ra + rb) mod n
= [(a mod n) + (b mod n)] mod n
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16. Modular Arithmetic (cont’d)
 Examples for the above three properties
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17. Modular Arithmetic (cont’d)
Properties of modular arithmetic
Let Zn = {0,1,2,…,(n-1)} be the set of residues modulo n.
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18. Modular Arithmetic (cont’d)
Properties of modular arithmetic (cont’d)
if (a + b)  (a + c) mod n, then b  c mod n (due to the existence of an additive inverse)
if (a  b)  (a  c) mod n, then b  c mod n (only if a is relatively prime to n; due to the possible absence of a multiplicative inverse)
e.g. 6  3 = 18  2 mod 8 and
6  7 = 42  2 mod 8 but
3  7 mod 8 (6 is not relatively prime to 8)
If n is prime then the property of multiplicative inverse holds (from a ring to a field).
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19. Modular Arithmetic (cont’d)
Properties of modular arithmetic (cont’d)
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20. Fermat’s and Euler’s Theorems
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21. Fermat’s and Euler’s Theorems (cont’d)
Fermat’s theorem (cont’d)
alternative form
if p is prime and a is any positive integer, then
ap  a mod p
example: p = 5, a = 3, 35 = 243  3 mod 5
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22. Fermat’s and Euler’s Theorems (cont’d)
Euler’s totient function
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23. Fermat’s and Euler’s Theorems (cont’d)
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24. Fermat’s and Euler’s Theorems (cont’d)
Euler’s totient function (cont’d)
if n is the product of two primes p and q
φ(n) = pq – [(q – 1)+(p –1) + 1]
= pq – (p + q) + 1
= (p – 1)  (q – 1)
= φ (p)  φ (q)
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25. Fermat’s and Euler’s Theorems (cont’d)
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26. Fermat’s and Euler’s Theorems (cont’d)
Euler’s totient function (cont’d)
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27. Information Security -- Public-Key Cryptography
Testing for Primality
If p is an odd prime, then the equation
x2  1 (mod p) has only two solutions, 1 and -1.
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28. Testing for Primality (cont’d)
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29. Testing for Primality (cont’d)
Probabilistic primality test
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30. Information Security -- Public-Key Cryptography
Euclid’s Algorithm
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31. Euclid’s Algorithm (cont’d)
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32. Euclid’s Algorithm (cont’d)
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33. Euclid’s Algorithm (cont’d)
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34. Extended Euclid’s Algorithm
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35. Chinese Remainder Theorem
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36. Chinese Remainder Theorem (cont’d)
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37. Information Security -- Public-Key Cryptography
Discrete Logarithms
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38. Discrete Logarithms (cont’d)
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