40 Theorems, Examples, RSA

40 Theorems, Examples, RSA - Handout#40 CS103A Robert...

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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
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2 Theorem 40.3 . If p is prime and n 1, then Φ (p n ) = p n – p n-1 . Examples: Consider p = 3 and n = 2.
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40 Theorems, Examples, RSA - Handout#40 CS103A Robert...

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