feat: add solovay strassen primality test (#5692)

* feat: add solovay strassen primality test

* chore: add wikipedia link

* fix: format and coverage

* fix: mvn stylecheck

* fix: PMD errors

* refactor: make random final

---------

Co-authored-by: Alex Klymenko <alexanderklmn@gmail.com>

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

@@ -0,0 +1,133 @@
+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.
+ *
+ * For more information, go to {@link https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test}
+ */
+final class SolovayStrassenPrimalityTest {
+
+    private final 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 = Math.abs(random.nextLong() % (num - 1)) + 2; // 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.
+    }
+}