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Euler's phi(n) function |
The Euler's phi(n) function is used to create the private key.
The Euler's phi(n) function is automatically calculated when both prime numbers p and q are available: phi(n) = (p - 1) * (q - 1) p = prime number p q = prime number q n = modulus (p * q) The Euler's phi function is defined as follows: phi(i) = the number of positive integers less than or equal to i that are coprime to i. Example 1:phi(8) = 4 Because 8 is coprime to 1, 3, 5 and 7 Example 2:phi(15) = 8 Because 15 is coprime to 1, 2, 4, 7, 8, 11, 13 and 14 If p is a prime number, then: phi(p) = (p-1) Example 3:phi(7) = 6 Because 7 is coprime to 1, 2, 3, 4, 5 and 6 Example 4:phi(13) = 12 Because 13 is coprime to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 If p and q are different prime numbers, then phi(p*q) = (p - 1)*(q - 1) Example 5:p=3, q=5 phi(3*5) = (3 - 1)*(5 - 1) phi(15) = 2*4 = 8 Because 15 is coprime to 1, 2, 4, 7, 8, 11, 13 and 14 Example 6:p=5, q=7 phi(5*7) = (5 - 1)*(7 - 1) phi(35) = 4*6 = 24 Because 35 is coprime to 1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33 and 34 |