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51 lines
1.2 KiB
Python
51 lines
1.2 KiB
Python
"""
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* Binary Exponentiation for Powers
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* This is a method to find a^b in a time complexity of O(log b)
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* This is one of the most commonly used methods of finding powers.
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* Also useful in cases where solution to (a^b)%c is required,
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* where a,b,c can be numbers over the computers calculation limits.
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* Done using iteration, can also be done using recursion
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* @author chinmoy159
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* @version 1.0 dated 10/08/2017
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"""
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def b_expo(a, b):
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res = 1
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while b > 0:
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if b & 1:
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res *= a
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a *= a
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b >>= 1
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return res
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def b_expo_mod(a, b, c):
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res = 1
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while b > 0:
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if b & 1:
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res = ((res % c) * (a % c)) % c
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a *= a
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b >>= 1
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return res
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"""
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* Wondering how this method works !
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* It's pretty simple.
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* Let's say you need to calculate a ^ b
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* RULE 1 : a ^ b = (a*a) ^ (b/2) ---- example : 4 ^ 4 = (4*4) ^ (4/2) = 16 ^ 2
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* RULE 2 : IF b is ODD, then ---- a ^ b = a * (a ^ (b - 1)) :: where (b - 1) is even.
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* Once b is even, repeat the process to get a ^ b
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* Repeat the process till b = 1 OR b = 0, because a^1 = a AND a^0 = 1
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*
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* As far as the modulo is concerned,
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* the fact : (a*b) % c = ((a%c) * (b%c)) % c
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* Now apply RULE 1 OR 2 whichever is required.
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"""
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