Add fast exponentiation algorithm (#5715)

src/main/java/com/thealgorithms/maths/FastExponentiation.java

@@ -0,0 +1,67 @@
+package com.thealgorithms.maths;
+
+/**
+ * This class provides a method to perform fast exponentiation (exponentiation by squaring),
+ * which calculates (base^exp) % mod efficiently.
+ *
+ * <p>The algorithm works by repeatedly squaring the base and reducing the exponent
+ * by half at each step. It exploits the fact that:
+ * <ul>
+ *   <li>If exp is even, (base^exp) = (base^(exp/2))^2</li>
+ *   <li>If exp is odd, (base^exp) = base * (base^(exp-1))</li>
+ * </ul>
+ * The result is computed modulo `mod` at each step to avoid overflow and keep the result within bounds.
+ * </p>
+ *
+ * <p><strong>Time complexity:</strong> O(log(exp)) — much faster than naive exponentiation (O(exp)).</p>
+ *
+ * For more information, please visit {@link https://en.wikipedia.org/wiki/Exponentiation_by_squaring}
+ */
+public final class FastExponentiation {
+
+    /**
+     * Private constructor to hide the implicit public one.
+     */
+    private FastExponentiation() {
+    }
+
+    /**
+     * Performs fast exponentiation to calculate (base^exp) % mod using the method
+     * of exponentiation by squaring.
+     *
+     * <p>This method efficiently computes the result by squaring the base and halving
+     * the exponent at each step. It multiplies the base to the result when the exponent is odd.
+     *
+     * @param base the base number to be raised to the power of exp
+     * @param exp the exponent to which the base is raised
+     * @param mod the modulus to ensure the result does not overflow
+     * @return (base^exp) % mod
+     * @throws IllegalArgumentException if the modulus is less than or equal to 0
+     * @throws ArithmeticException if the exponent is negative (not supported in this implementation)
+     */
+    public static long fastExponentiation(long base, long exp, long mod) {
+        if (mod <= 0) {
+            throw new IllegalArgumentException("Modulus must be positive.");
+        }
+
+        if (exp < 0) {
+            throw new ArithmeticException("Negative exponent is not supported.");
+        }
+
+        long result = 1;
+        base = base % mod; // Take the modulus of the base to handle large base values
+
+        // Fast exponentiation by squaring algorithm
+        while (exp > 0) {
+            // If exp is odd, multiply the base to the result
+            if ((exp & 1) == 1) { // exp & 1 checks if exp is odd
+                result = result * base % mod;
+            }
+            // Square the base and halve the exponent
+            base = base * base % mod; // base^2 % mod to avoid overflow
+            exp >>= 1; // Right shift exp to divide it by 2
+        }
+
+        return result;
+    }
+}