Handout #40
CS103A
February 20, 2008
Robert Plummer
Number Theory: Theorems, Examples, and RSA
This handout will list and give examples for some key theorems of Number Theory, and
it will provide details about the RSA public-key encryption algorithm.
Preliminaries
The Euler
Φ
function
is defined as follows:
Φ
(n) is the number of positive integers less
than n that are relatively prime to n.
This function just counts how many without concern for what the values are.
For
example,
Φ
(8) = 4, since the positive numbers less than 8 that are relatively prime to it
are {1, 3, 5, 7}.
Here are some additional examples:
Φ
(3) =
2
since {1, 2} are relatively prime to 3
Φ
(5) =
4
since {1, 2, 3, 4} are relatively prime to 5
Φ
(10) = 4
since {1, 3, 7, 9} are relatively prime to 10
Φ
(15) = 8
since {1, 2, 4, 7, 8, 11, 13, 14} are relatively prime to 15
Many interesting facts can be proved about the Euler
Φ
function, including:
Theorem 40.1
.
If p is prime then
Φ
(p) = p – 1.
Examples:
Φ
(3) and
Φ
(5) above.
Theorem 40.2
.
If gcd(n, m) = 1, then
Φ
(nm) =
Φ
(n) ·
Φ
(m).
Recall that gcd(n, m) = 1 is another way of saying that n and m are relatively prime.
Another way to say this is that n and m are coprime.
Examples: 3 and 5 are relatively prime, so the theorem claims that
Φ
(15) =
Φ
(3) ·
Φ
(5) = 2 · 4 = 8, which is correct.
Also, we can find
Φ
(30) to be 8 by inspection (underlined items are relatively prime to
30):
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
and
Φ
(3) ·
Φ
(10) = 2 · 4 = 8