Euler_Phi and Its Applications

Euler_Phi, the LCM method and the Squaring method

Euler_Phi.zip is a suite of the foll Programmes :

1) Euler_Phi (n) returns the no of positive integers less than n which are prime to n.

2) a_k_mod_m_LCM_Method (a, m, k) : Imagine computing mod(14^26, 45) or mod(56^3005, 1125).
This programme computes a_k_mod_m = mod (a^k, m) when a and m are coprime, using certain rules for the reduction of computations based on Euler_Phi.

3) b_n_mod_m_Rpt_Sq_Method (b, n, m) : Imagine computing mod(38^75, 103) or mod(38^75, 103). This programme computes b_n_mod_m = mod (b^n, m).
The diff between this pgm and a_k_mod_m_LCM_Method.m is that in a_k_mod_m_LCM_Method,
the inputs a and m are coprime ; here, in b_n_mod_m_Rpt_Sq_Method.m, it is not necessary that b and m need be coprime.

4) Verify_Euler_Phi(n) : This is a small programme to prove that n = sum of Euler_Phi taken over all divisors d of n.

Refer to Euler_Phi function in P15, P22, P23, Prob 21 & Prob 23 / (P26 + P205) in the book : A course in Number Theory and Cryptography by Neal Koblitz.
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