Solovay-Strassen method of Primality Test

Last Updated : 11 Feb, 2025

We have already been introduced to primality testing in the previous articles in this series.

The Solovay–Strassen test is a probabilistic algorithm used to check if a number is prime or composite (not prime). It is not 100% accurate but gives a high probability of correctness when run multiple times. Before jumping into the test, let's break down some key concepts:

To perform the test, we need to understand two important mathematical symbols:

Legendre Symbol (a/p)
Jacobian Symbol (a/n)

These symbols help us determine properties of numbers in a structured way.

Legendre Symbol (a/p)

This is a special way to compare two numbers:
a: Any integer
p: A prime number
It is written as (a/p) and tells us one of the following three things:
(a/p) = 0 → If a is divisible by p, meaning a % p == 0
(a/p) = 1 → if there exists an integer k such that k²= a(mod p)
(a/p) = -1 → Otherwise (if the above two cases are false)
Example
If p = 7, and a = 9, then:
(9/7) = 1 because 9 has a square root 3 (since 3² ≡ 9).
If a = 2, then (2/7) = -1 because 2 is not a square modulo 7.
Mathematician Euler discovered a useful shortcut to compute (a/p):
(a/p) = a^(p-1)/2 mod p
This formula helps in quickly checking the Legendre symbol.

Jacobian Symbol (a/n)

The Jacobian symbol is a generalization of the Legendre symbol, but it works when n is not necessarily a prime. It is calculated by breaking down n into prime factors and applying the Legendre symbol to each factor.
If n is a prime number, then:
(a/n) = (a/p)
So, Jacobian symbol = Legendre symbol when n is prime.

The Solovay–Strassen Primality Test

Now that we understand these symbols, let's go step by step through the primality test.

Step 1: Choose a Random Number
Pick a random number a, such that 2 ≤ a ≤ n - 1.
This number a acts as a "witness" to help determine whether n is prime.
Step 2: Check the GCD (Greatest Common Divisor)
Compute gcd(a, n) (the largest number that divides both a and n).
If gcd(a, n) > 1, then n is definitely composite (not prime).
This step ensures n has no common factors with a.
Step 3: Compute Two Values
We compute two important values:
a^((n-1)/2) mod n (using fast exponentiation)
(a/n) (using the Jacobian symbol)
Step 4: Compare the Two Values
If these two values do not match, then n is composite.
If they are equal, then n is probably prime.

Why Do We Repeat This Test Multiple Times?

Since this is a probabilistic test, we cannot be 100% sure that n is prime after one test. So, we repeat the test with different random values of a multiple times.

If the test fails even once, n is definitely composite.
If the test passes many times, then n is probably prime.

C++

// C++ program to implement Solovay-Strassen Primality Test #include <bits/stdc++.h> using namespace std;  // Function to perform modular exponentiation long long modulo(long long base, long long exponent, long long mod) {     long long x = 1, y = base;      while (exponent > 0) {         if (exponent % 2 == 1) {             x = (x * y) % mod;         }         y = (y * y) % mod;         exponent /= 2;     }     return x % mod; }  // Function to calculate the Jacobian symbol (a/n) int calculateJacobian(long long a, long long n) {     if (a == 0) {         return 0; // (0/n) = 0     }      int ans = 1;      if (a < 0) {         a = -a; // (a/n) = (-a/n) * (-1/n)         if (n % 4 == 3) {             ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)         }     }      if (a == 1) {         return ans; // (1/n) = 1     }      while (a) {         if (a < 0) {             a = -a; // (a/n) = (-a/n) * (-1/n)             if (n % 4 == 3) {                 ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)             }         }          while (a % 2 == 0) {             a /= 2;             if (n % 8 == 3 || n % 8 == 5) {                 ans = -ans;             }         }          swap(a, n);          if (a % 4 == 3 && n % 4 == 3) {             ans = -ans;         }          a %= n;          if (a > n / 2) {             a -= n;         }     }      return (n == 1) ? ans : 0; }  // Function to perform the Solovay-Strassen Primality Test bool solovoyStrassen(long long p, int iterations) {     if (p < 2 || (p % 2 == 0 && p != 2)) {         return false;     }      for (int i = 0; i < iterations; i++) {         long long a = rand() % (p - 1) + 1;         long long jacobian = (p + calculateJacobian(a, p)) % p;         long long mod = modulo(a, (p - 1) / 2, p);          if (!jacobian || mod != jacobian) {             return false;         }     }     return true; }  int main() {     int iterations = 50;     long long num1 = 15, num2 = 13;      if (solovoyStrassen(num1, iterations)) {         printf("%lld is prime\n", num1);     } else {         printf("%lld is composite\n", num1);     }      if (solovoyStrassen(num2, iterations)) {         printf("%lld is prime\n", num2);     } else {         printf("%lld is composite\n", num2);     }      return 0; }

Java

// Java program to implement Solovay-Strassen Primality Test  import java.util.Random;  class GFG {          // Modulo function to perform binary exponentiation     static long modulo(long base, long exponent, long mod) {         long x = 1;         long y = base;          while (exponent > 0) {             if (exponent % 2 == 1) {                 x = (x * y) % mod;             }             y = (y * y) % mod;             exponent /= 2;         }         return x % mod;     }      // Function to calculate the Jacobian symbol of a given number     static long calculateJacobian(long a, long n) {         if (n <= 0 || n % 2 == 0) {             return 0;         }          long ans = 1;         if (a < 0) {             a = -a; // (a/n) = (-a/n) * (-1/n)             if (n % 4 == 3) {                 ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)             }         }         if (a == 1) {             return ans; // (1/n) = 1         }          while (a != 0) {             if (a < 0) {                 a = -a; // (a/n) = (-a/n) * (-1/n)                 if (n % 4 == 3) {                     ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)                 }             }             while (a % 2 == 0) {                 a /= 2;                 if (n % 8 == 3 || n % 8 == 5) {                     ans = -ans;                 }             }              long temp = a;             a = n;             n = temp;              if (a % 4 == 3 && n % 4 == 3) {                 ans = -ans;             }             a %= n;             if (a > n / 2) {                 a -= n;             }         }         return (n == 1) ? ans : 0;     }      // Function to perform the Solovay-Strassen Primality Test     static boolean solovayStrassen(long p, int iteration) {         if (p < 2) {             return false;         }         if (p != 2 && p % 2 == 0) {             return false;         }          Random rand = new Random();         for (int i = 0; i < iteration; i++) {             long r = Math.abs(rand.nextLong());             long a = r % (p - 1) + 1;             long jacobian = (p + calculateJacobian(a, p)) % p;             long mod = modulo(a, (p - 1) / 2, p);              if (jacobian == 0 || mod != jacobian) {                 return false;             }         }         return true;     }      // Driver code     public static void main(String[] args) {         int iter = 50;         long num1 = 15;         long num2 = 13;          if (solovayStrassen(num1, iter)) {             System.out.println(num1 + " is prime");         } else {             System.out.println(num1 + " is composite");         }          if (solovayStrassen(num2, iter)) {             System.out.println(num2 + " is prime");         } else {             System.out.println(num2 + " is composite");         }     } }

Python

# Python3 program to implement Solovay-Strassen  # Primality Test  import random  # modulo function to perform binary exponentiation def modulo(base, exponent, mod):     x = 1     y = base     while exponent > 0:         if exponent % 2 == 1:             x = (x * y) % mod          y = (y * y) % mod         exponent //= 2      return x % mod  # To calculate Jacobian symbol of a given number def calculateJacobian(a, n):     if a == 0:         return 0  # (0/n) = 0      ans = 1     if a < 0:         # (a/n) = (-a/n)*(-1/n)         a = -a         if n % 4 == 3:             # (-1/n) = -1 if n = 3 (mod 4)             ans = -ans      if a == 1:         return ans  # (1/n) = 1      while a:         if a < 0:             # (a/n) = (-a/n)*(-1/n)             a = -a             if n % 4 == 3:                 # (-1/n) = -1 if n = 3 (mod 4)                 ans = -ans          while a % 2 == 0:             a //= 2             if n % 8 == 3 or n % 8 == 5:                 ans = -ans          # Swap         a, n = n, a          if a % 4 == 3 and n % 4 == 3:             ans = -ans         a %= n          if a > n // 2:             a -= n      if n == 1:         return ans      return 0  # To perform the Solovay-Strassen Primality Test def solovoyStrassen(p, iterations):     if p < 2:         return False     if p != 2 and p % 2 == 0:         return False      for _ in range(iterations):         # Generate a random number a         a = random.randrange(1, p)         jacobian = (p + calculateJacobian(a, p)) % p         mod = modulo(a, (p - 1) // 2, p)          if jacobian == 0 or mod != jacobian:             return False      return True  if __name__ == "__main__":     iterations = 50     num1 = 15     num2 = 13      if solovoyStrassen(num1, iterations):         print(num1, "is prime")     else:         print(num1, "is composite")      if solovoyStrassen(num2, iterations):         print(num2, "is prime")     else:         print(num2, "is composite")

// C# program to implement the Solovay-Strassen Primality Test  using System;  class GfG {     // Function to perform modular exponentiation     static long Modulo(long Base, long exponent, long mod) {         long x = 1;         long y = Base;          while (exponent > 0) {             if (exponent % 2 == 1) {                 x = (x * y) % mod;             }             y = (y * y) % mod;             exponent /= 2;         }         return x % mod;     }      // Function to calculate the Jacobian symbol of a given number     static long CalculateJacobian(long a, long n) {         if (n <= 0 || n % 2 == 0) {             return 0;         }          long ans = 1;         if (a < 0) {             a = -a; // (a/n) = (-a/n) * (-1/n)             if (n % 4 == 3) {                 ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)             }         }         if (a == 1) {             return ans; // (1/n) = 1         }          while (a != 0) {             if (a < 0) {                 a = -a; // (a/n) = (-a/n) * (-1/n)                 if (n % 4 == 3) {                     ans = -ans; // (-1/n) = -1 if n ≡ 3 (mod 4)                 }             }             while (a % 2 == 0) {                 a /= 2;                 if (n % 8 == 3 || n % 8 == 5) {                     ans = -ans;                 }             }              long temp = a;             a = n;             n = temp;              if (a % 4 == 3 && n % 4 == 3) {                 ans = -ans;             }             a %= n;             if (a > n / 2) {                 a -= n;             }         }         return (n == 1) ? ans : 0;     }      // Function to perform the Solovay-Strassen Primality Test     static bool SolovayStrassen(long p, int iteration) {         if (p < 2) {             return false;         }         if (p != 2 && p % 2 == 0) {             return false;         }          Random rand = new Random();         for (int i = 0; i < iteration; i++) {             long r = Math.Abs(rand.Next());             long a = r % (p - 1) + 1;             long jacobian = (p + CalculateJacobian(a, p)) % p;             long mod = Modulo(a, (p - 1) / 2, p);              if (jacobian == 0 || mod != jacobian) {                 return false;             }         }         return true;     }      // Driver Code     static void Main() {         int iter = 50;         long num1 = 15;         long num2 = 13;          if (SolovayStrassen(num1, iter)) {             Console.WriteLine(num1 + " is prime");         } else {             Console.WriteLine(num1 + " is composite");         }          if (SolovayStrassen(num2, iter)) {             Console.WriteLine(num2 + " is prime");         } else {             Console.WriteLine(num2 + " is composite");         }     } }

JavaScript

// JavaScript program to implement Solovay-Strassen Primality Test  // Modulo function to perform binary exponentiation function modulo(base, exponent, mod) {     let x = 1;     let y = base;          while (exponent > 0) {         if (exponent % 2 == 1) {             x = (x * y) % mod;         }         y = (y * y) % mod;         exponent = Math.floor(exponent / 2);     }     return x % mod; }  // To calculate Jacobian symbol of a given number function calculateJacobian(a, n) {     if (n <= 0 || n % 2 == 0) {         return 0;     }          let ans = 1;     if (a < 0) {         a = -a;         if (n % 4 == 3) {             ans = -ans;         }     }          if (a == 1) {         return ans;     }          while (a !== 0) {         if (a < 0) {             a = -a;             if (n % 4 == 3) {                 ans = -ans;             }         }                  while (a % 2 == 0) {             a = Math.floor(a / 2);             if (n % 8 == 3 || n % 8 == 5) {                 ans = -ans;             }         }                  let temp = a;         a = n;         n = temp;                  if (a % 4 == 3 && n % 4 == 3) {             ans = -ans;         }                  a %= n;         if (a > Math.floor(n / 2)) {             a -= n;         }     }          return n === 1 ? ans : 0; }  // To perform the Solovay-Strassen Primality Test function solovoyStrassen(p, iteration) {     if (p < 2 || (p % 2 === 0 && p !== 2)) {         return false;     }          for (let i = 0; i < iteration; i++) {         let a = Math.floor(Math.random() * (p - 2)) + 1;         let jacobian = (p + calculateJacobian(a, p)) % p;         let mod = modulo(a, Math.floor((p - 1) / 2), p);                  if (jacobian === 0 || mod !== jacobian) {             return false;         }     }     return true; }  // Driver Code let iter = 50; let num1 = 15; let num2 = 13;  console.log(num1 + (solovoyStrassen(num1, iter) ? " is prime" : " is composite")); console.log(num2 + (solovoyStrassen(num2, iter) ? " is prime" : " is composite"));

Output :

15 is composite
13 is prime

Time Complexity: Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k*(log n)³), where k is the number of different values we test.
Auxiliary Space: O(1) as it is using constant space for variables

Accuracy: It is possible for the algorithm to return an incorrect answer. If the input n is indeed prime, then the output will always probably be correctly prime. However, if the input n is composite, then it is possible for the output to probably be incorrect prime. The number n is then called an Euler-Jacobi pseudoprime.

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Solovay-Strassen method of Primality Test

Legendre Symbol (a/p)

Example

Jacobian Symbol (a/n)

The Solovay–Strassen Primality Test

Step 1: Choose a Random Number

Step 2: Check the GCD (Greatest Common Divisor)

Step 3: Compute Two Values

Step 4: Compare the Two Values

Why Do We Repeat This Test Multiple Times?

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