feat: add solovay strassen primality test

Author: saahil-mahato

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

@@ -0,0 +1,126 @@
+package com.thealgorithms.maths;
+
+import java.util.Random;
+
+/**
+ * This class implements the Solovay-Strassen primality test, 
+ * which is a probabilistic algorithm to determine whether a number is prime.
+ * The algorithm is based on properties of the Jacobi symbol and modular exponentiation.
+ */
+class SolovayStrassenPrimalityTest {
+
+    private Random random;
+
+    /**
+     * Constructs a SolovayStrassenPrimalityTest instance with a specified seed for randomness.
+     *
+     * @param seed the seed for generating random numbers
+     */
+    private SolovayStrassenPrimalityTest(int seed) {
+        random = new Random(seed);
+    }
+
+    /**
+     * Factory method to create an instance of SolovayStrassenPrimalityTest.
+     *
+     * @param seed the seed for generating random numbers
+     * @return a new instance of SolovayStrassenPrimalityTest
+     */
+    public static SolovayStrassenPrimalityTest getSolovayStrassenPrimalityTest(int seed) {
+        return new SolovayStrassenPrimalityTest(seed);
+    }
+
+    /**
+     * Calculates modular exponentiation using the method of exponentiation by squaring.
+     *
+     * @param base the base number
+     * @param exponent the exponent
+     * @param mod the modulus
+     * @return (base^exponent) mod mod
+     */
+    private static long calculateModularExponentiation(long base, long exponent, long mod) {
+        long x = 1; // This will hold the result of (base^exponent) % mod
+        long y = base; // This holds the current base value being squared
+
+        while (exponent > 0) {
+            // If exponent is odd, multiply the current base (y) with x
+            if (exponent % 2 == 1)
+                x = (x * y) % mod; // Update result with current base
+
+            // Square the base for the next iteration
+            y = (y * y) % mod; // Update base to be y^2
+            exponent = exponent / 2; // Halve the exponent for next iteration
+        }
+        
+        return x % mod; // Return final result after all iterations
+    }
+
+    /**
+     * Computes the Jacobi symbol (a/n), which is a generalization of the Legendre symbol.
+     *
+     * @param a the numerator
+     * @param num the denominator (must be an odd positive integer)
+     * @return the Jacobi symbol value: 1, -1, or 0
+     */
+    public int calculateJacobi(long a, long num) {
+        // Check if num is non-positive or even; Jacobi symbol is not defined in these cases
+        if (num <= 0 || num % 2 == 0) {
+            return 0;
+        }
+
+        a = a % num; // Reduce a modulo num to simplify calculations
+        int jacobi = 1; // Initialize Jacobi symbol value
+
+        while (a != 0) {
+            // While a is even, reduce it and adjust jacobi based on properties of num
+            while (a % 2 == 0) {
+                a /= 2; // Divide a by 2 until it becomes odd
+                long nMod8 = num % 8; // Get num modulo 8 to check conditions for jacobi adjustment
+                if (nMod8 == 3 || nMod8 == 5) {
+                    jacobi = -jacobi; // Flip jacobi sign based on properties of num modulo 8
+                }
+            }
+
+            long temp = a; // Temporarily store value of a
+            a = num; // Set a to be num for next iteration
+            num = temp; // Set num to be previous value of a
+
+            // Adjust jacobi based on properties of both numbers when both are odd and congruent to 3 modulo 4
+            if (a % 4 == 3 && num % 4 == 3) {
+                jacobi = -jacobi; // Flip jacobi sign again based on congruences
+            }
+
+            a = a % num; // Reduce a modulo num for next iteration of Jacobi computation
+        }
+
+        return (num == 1) ? jacobi : 0; // If num reduces to 1, return jacobi value, otherwise return 0 (not defined)
+    }
+
+    /**
+     * Performs the Solovay-Strassen primality test on a given number.
+     *
+     * @param num the number to be tested for primality
+     * @param iterations the number of iterations to run for accuracy
+     * @return true if num is likely prime, false if it is composite
+     */
+    public boolean solovayStrassen(long num, int iterations) {
+        if (num <= 1) return false; // Numbers <=1 are not prime by definition.
+        if (num <= 3) return true;   // Numbers <=3 are prime.
+
+        for (int i = 0; i < iterations; i++) {
+            long r = (random.nextLong() & Long.MAX_VALUE); // Generate a non-negative random number.
+            long a = r % (num - 1) + 1; // Choose random 'a' in range [1, n-1].
+            
+            long jacobi = (num + calculateJacobi(a, num)) % num; 
+            // Calculate Jacobi symbol and adjust it modulo n. 
+            
+            long mod = calculateModularExponentiation(a, (num - 1) / 2, num); 
+            // Calculate modular exponentiation: a^((n-1)/2) mod n.
+
+            if (jacobi == 0 || mod != jacobi) return false; 
+            // If Jacobi symbol is zero or doesn't match modular result, n is composite.
+        }
+
+        return true; // If no contradictions found after all iterations, n is likely prime.
+    }
+}

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

@@ -0,0 +1,130 @@
+package com.thealgorithms.maths;
+
+import org.junit.jupiter.api.BeforeEach;
+import org.junit.jupiter.api.Test;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.MethodSource;
+
+import static org.junit.jupiter.api.Assertions.*;
+
+/**
+ * Unit tests for the {@link SolovayStrassenPrimalityTest} class.
+ * This class tests the functionality of the Solovay-Strassen primality test implementation.
+ */
+class SolovayStrassenPrimalityTestTest {
+
+    private static final int RANDOM_SEED = 123; // Seed for reproducibility
+    private SolovayStrassenPrimalityTest testInstance;
+
+    /**
+     * Sets up a new instance of {@link SolovayStrassenPrimalityTest} 
+     * before each test case, using a fixed random seed for consistency.
+     */
+    @BeforeEach
+    void setUp() {
+        testInstance = SolovayStrassenPrimalityTest.getSolovayStrassenPrimalityTest(RANDOM_SEED);
+    }
+
+    /**
+     * Provides test cases for prime numbers with various values of n and k (iterations).
+     *
+     * @return an array of objects containing pairs of n and k values
+     */
+    static Object[][] primeNumbers() {
+        return new Object[][]{
+                {2, 1}, {3, 1}, {5, 5}, {7, 10},
+                {11, 20}, {13, 10}, {17, 5}, {19, 1}
+        };
+    }
+
+    /**
+     * Tests known prime numbers with various values of n and k (iterations).
+     *
+     * @param n the number to be tested for primality
+     * @param k the number of iterations to use in the primality test
+     */
+    @ParameterizedTest
+    @MethodSource("primeNumbers")
+    void testPrimeNumbersWithDifferentNAndK(int n, int k) {
+        assertTrue(testInstance.solovayStrassen(n, k), n + " should be prime");
+    }
+
+    /**
+     * Provides test cases for composite numbers with various values of n and k (iterations).
+     *
+     * @return an array of objects containing pairs of n and k values
+     */
+    static Object[][] compositeNumbers() {
+        return new Object[][]{
+                {4, 1}, {6, 5}, {8, 10}, {9, 20},
+                {10, 1}, {12, 5}, {15, 10}
+        };
+    }
+
+    /**
+     * Tests known composite numbers with various values of n and k (iterations).
+     *
+     * @param n the number to be tested for primality
+     * @param k the number of iterations to use in the primality test
+     */
+    @ParameterizedTest
+    @MethodSource("compositeNumbers")
+    void testCompositeNumbersWithDifferentNAndK(int n, int k) {
+        assertFalse(testInstance.solovayStrassen(n, k), n + " should be composite");
+    }
+
+    /**
+     * Tests edge cases for the primality test.
+     * This includes negative numbers and small integers (0 and 1).
+     */
+    @Test
+    void testEdgeCases() {
+        assertFalse(testInstance.solovayStrassen(-1, 10), "-1 should not be prime");
+        assertFalse(testInstance.solovayStrassen(0, 10), "0 should not be prime");
+        assertFalse(testInstance.solovayStrassen(1, 10), "1 should not be prime");
+
+        // Test small primes and composites
+        assertTrue(testInstance.solovayStrassen(2, 1), "2 is a prime number (single iteration)");
+        assertFalse(testInstance.solovayStrassen(9, 1), "9 is a composite number (single iteration)");
+
+        // Test larger primes and composites
+        long largePrime = 104729; // Known large prime number
+        long largeComposite = 104730; // Known composite number (even)
+        
+        assertTrue(testInstance.solovayStrassen(largePrime, 20), "104729 is a prime number");
+        assertFalse(testInstance.solovayStrassen(largeComposite, 20), "104730 is a composite number");
+
+        // Test very large numbers (may take longer)
+        long veryLargePrime = 512927357; // Near the upper limit of long
+        long veryLargeComposite = 512927358; // Composite (even)
+        
+        assertTrue(testInstance.solovayStrassen(veryLargePrime, 20), 
+                   Long.MAX_VALUE - 1 + " is likely a prime number.");
+        
+        assertFalse(testInstance.solovayStrassen(veryLargeComposite, 20),
+                    Long.MAX_VALUE + " is a composite number.");
+    }
+
+    /**
+     * Tests the Jacobi symbol calculation directly for known values.
+     * This verifies that the Jacobi symbol method behaves as expected.
+     */
+    @Test
+    void testJacobiSymbolCalculation() {
+        // Jacobi symbol (a/n) where n is odd and positive
+        int jacobi1 = testInstance.calculateJacobi(6, 11); // Should return -1
+        int jacobi2 = testInstance.calculateJacobi(5, 11); // Should return +1
+        int jacobi3 = testInstance.calculateJacobi(4, 11); // Should return +1
+
+        assertEquals(-1, jacobi1);
+        assertEquals(+1, jacobi2);
+        assertEquals(+1, jacobi3);
+        
+        // Edge case: Jacobi symbol with even n or non-positive n
+        int jacobi4 = testInstance.calculateJacobi(5, -11); // Should return 0 (invalid)
+        int jacobi5 = testInstance.calculateJacobi(5, 0);   // Should return 0 (invalid)
+        
+        assertEquals(0, jacobi4);
+        assertEquals(0, jacobi5);
+    }
+}