This document discusses Fermat's theorem and Euler's theorem, which are related theorems about integers modulo prime numbers. It provides proofs of both theorems and explains their applications. It also defines Euler's totient function φ(n) and provides examples to demonstrate how it can be used to calculate φ(n) for composite numbers.

1. FERMAT AND
EULER THEOREM
~ S. Janani, AP/CSE
KCET

2. Contents
• Fermats Theorem
• Proof
• Eulers Totient Function
• Eulers Theorem
• Proof
• Applications
• Example

3. Fermats Theorem



4. Proof
 Consider a set of positive integers less than ‘p’ :
• {1,2,3,…..,(p-1)} and multiply each element by ‘a’ and
‘modulo p’ , to get the set
• X = {a mod p, 2a mod p,…, (p-1)a mod p}
 No elements of X is zero and equal, since p doesn’t
divide a.
 Multiplying the numbers in both sets (p and X) and
taking the result mod p yields

5. Proof (Contd.,)
• Multiplying the numbers in both sets (p and X) and taking
the result mod p yields
• a * 2a *…* (p-1)a  [1 * 2 * 3 *…* (p-1)] (mod p)
• a p  1
( p  1 ) !  ( p  1) ! ( mod p)
• Thus on equating (p-1)! term from both the sides, since it
is relatively prime to p, result becomes,
An alternative form of Fermat’s Theorem is given as
 1 ( m o d p)
a p
 a ( m o d p )
a p  1

6. EULER TOTIENT FUNCTION : φ (n)
  (n) : How many numbers there are between 1 and n-
1 that are relatively prime to n.
  (4) = 2 (1, 3 are relatively prime to 4).
  (5) = 4 (1, 2, 3, 4 are relatively prime to 5).
  (6) = 2 (1, 5 are relatively prime to 6).
  (7) = 6 (1, 2, 3, 4, 5, 6 are relatively prime to 7).

7. EULER TOTIENT FUNCTION : φ (n)
 From  (5) and  (7), (n) will be n-1
whenever n is a prime number.
 This implies that  (n) will be easy to calculate when n
has exactly two different prime factors:
(P * Q) = (P-1)*(Q-1)
if P and Q are prime.

8. Eulers Theorem
a n  1mod n

9. Above equation is true if n is prime because then,
and Fermat’s theorem holds.
Consider the set of such integers, labeled as,
Here each element xi of R is unique positive integer less than n
with GCD( xi ,n ) = 1.
n  ( n 1)

10. Proof
• Multiply each element by a, modulo n :
The set S is permutation of R :
• Because a and xi is relatively prime to n, so multiplication
is also be relatively prime to n. Thus the elements of S are
integers that are less than n and that are relatively prime
to n.
• There are no duplicates in S.

11. Proof

12. Applications

13. Example
• Solve 3 ^ 21 mod 11

18. Quiz
• Who am I
• I can scramble the text
• One key is not enough for me
• Its more secure if you exploit me
Asymmetric Key Encryption / Public key
encryption

19. • Who am I
• I was designed by 3 people
• Fermats theorem is my basis
• I can perform encryption
• I need more than one key for my operation
RSA Algorithm
Quiz

20. • Who am I
• I was designed by 2 people
• One key is not enough for me to do my operation
• Discrete Logarithm is my basis
• I am just used to share the personal data
• I can’t hold the identity
Diffie Hellman Key Exchange Algorithm
Quiz