feat: add fast exponentiation algorithm

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

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+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;
+    }
+}

src/test/java/com/thealgorithms/maths/FastExponentiationTest.java

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+package com.thealgorithms.maths;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+
+import org.junit.jupiter.api.Test;
+
+/**
+ * Unit tests for the {@link FastExponentiation} class.
+ *
+ * <p>This class contains various test cases to verify the correctness of the fastExponentiation method.
+ * It covers basic functionality, edge cases, and exceptional cases.
+ */
+class FastExponentiationTest {
+
+    /**
+     * Tests fast exponentiation with small numbers.
+     */
+    @Test
+    void testSmallNumbers() {
+        assertEquals(1024, FastExponentiation.fastExponentiation(2, 10, 10000), "2^10 mod 10000 should be 1024");
+        assertEquals(81, FastExponentiation.fastExponentiation(3, 4, 1000), "3^4 mod 1000 should be 81");
+    }
+
+    /**
+     * Tests the behavior of the fast exponentiation method when using a modulus.
+     */
+    @Test
+    void testWithModulo() {
+        assertEquals(24, FastExponentiation.fastExponentiation(2, 10, 1000), "2^10 mod 1000 should be 24");
+        assertEquals(0, FastExponentiation.fastExponentiation(10, 5, 10), "10^5 mod 10 should be 0");
+    }
+
+    /**
+     * Tests the edge cases where base or exponent is 0.
+     */
+    @Test
+    void testBaseCases() {
+        assertEquals(1, FastExponentiation.fastExponentiation(2, 0, 1000), "Any number raised to the power 0 mod anything should be 1");
+        assertEquals(0, FastExponentiation.fastExponentiation(0, 10, 1000), "0 raised to any power should be 0");
+        assertEquals(1, FastExponentiation.fastExponentiation(0, 0, 1000), "0^0 is considered 0 in modular arithmetic.");
+    }
+
+    /**
+     * Tests fast exponentiation with a negative base to ensure correctness under modular arithmetic.
+     */
+    @Test
+    void testNegativeBase() {
+        assertEquals(9765625, FastExponentiation.fastExponentiation(-5, 10, 1000000007), "-5^10 mod 1000000007 should be 9765625");
+    }
+
+    /**
+     * Tests that a negative exponent throws an ArithmeticException.
+     */
+    @Test
+    void testNegativeExponent() {
+        assertThrows(ArithmeticException.class, () -> { FastExponentiation.fastExponentiation(2, -5, 1000); });
+    }
+
+    /**
+     * Tests that the method throws an IllegalArgumentException for invalid modulus values.
+     */
+    @Test
+    void testInvalidModulus() {
+        assertThrows(IllegalArgumentException.class, () -> { FastExponentiation.fastExponentiation(2, 5, 0); });
+    }
+}