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68 lines
2.6 KiB
Java
68 lines
2.6 KiB
Java
package com.thealgorithms.maths;
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/**
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* This class provides a method to perform fast exponentiation (exponentiation by squaring),
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* which calculates (base^exp) % mod efficiently.
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*
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* <p>The algorithm works by repeatedly squaring the base and reducing the exponent
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* by half at each step. It exploits the fact that:
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* <ul>
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* <li>If exp is even, (base^exp) = (base^(exp/2))^2</li>
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* <li>If exp is odd, (base^exp) = base * (base^(exp-1))</li>
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* </ul>
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* The result is computed modulo `mod` at each step to avoid overflow and keep the result within bounds.
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* </p>
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*
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* <p><strong>Time complexity:</strong> O(log(exp)) — much faster than naive exponentiation (O(exp)).</p>
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*
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* For more information, please visit {@link https://en.wikipedia.org/wiki/Exponentiation_by_squaring}
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*/
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public final class FastExponentiation {
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/**
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* Private constructor to hide the implicit public one.
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*/
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private FastExponentiation() {
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}
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/**
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* Performs fast exponentiation to calculate (base^exp) % mod using the method
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* of exponentiation by squaring.
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*
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* <p>This method efficiently computes the result by squaring the base and halving
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* the exponent at each step. It multiplies the base to the result when the exponent is odd.
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*
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* @param base the base number to be raised to the power of exp
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* @param exp the exponent to which the base is raised
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* @param mod the modulus to ensure the result does not overflow
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* @return (base^exp) % mod
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* @throws IllegalArgumentException if the modulus is less than or equal to 0
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* @throws ArithmeticException if the exponent is negative (not supported in this implementation)
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*/
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public static long fastExponentiation(long base, long exp, long mod) {
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if (mod <= 0) {
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throw new IllegalArgumentException("Modulus must be positive.");
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}
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if (exp < 0) {
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throw new ArithmeticException("Negative exponent is not supported.");
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}
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long result = 1;
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base = base % mod; // Take the modulus of the base to handle large base values
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// Fast exponentiation by squaring algorithm
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while (exp > 0) {
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// If exp is odd, multiply the base to the result
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if ((exp & 1) == 1) { // exp & 1 checks if exp is odd
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result = result * base % mod;
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}
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// Square the base and halve the exponent
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base = base * base % mod; // base^2 % mod to avoid overflow
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exp >>= 1; // Right shift exp to divide it by 2
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}
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return result;
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}
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}
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