How to Efficiently Perform Large Number Primality Testing with PHP and GMP (Fermat’s Theorem)

Introduction to Fermat’s Theorem and Primality Testing

In number theory and cryptography, Fermat’s theorem is widely used for primality testing. Its core idea states: if a number $n$ is prime, then for any integer $a$ less than $n$ and coprime to it, the following must hold: $a^{n-1} \equiv 1 \mod n$. Based on this principle, we can construct a fast, probabilistic algorithm to test for primality.

What is the GMP Extension?

GMP (GNU Multiple Precision Arithmetic Library) is a high-performance library designed for arithmetic with large numbers. PHP supports GMP via an extension, which provides functions to perform various big integer operations like exponentiation, modulo, and random number generation.

To install GMP on a Linux system, use the following command:

sudo apt-get install php-gmp

On Windows, simply uncomment the line extension=php_gmp.dll in your php.ini file and restart your PHP service.

Implementing Fermat’s Primality Test in PHP

Below is a PHP function using GMP to implement Fermat’s test and determine the likelihood that a large number is prime:

function fermatTest($n, $k){
    if ($n == 2){
        return true; // 2 is a prime number
    }
    if ($n < 2 || $n % 2 == 0){
        return false; // Even numbers and numbers < 2 are not prime
    }
    for ($i = 0; $i < $k; $i++){
        $a = gmp_random_range(2, $n - 2); // Random number between 2 and $n-2
        $r = gmp_powm($a, $n - 1, $n); // Compute a^(n-1) mod n
        if (gmp_cmp($r, 1) != 0){
            return false; // Does not satisfy Fermat’s condition
        }
    }
    return true; // Likely a prime number
}

In this function, $n is the number to test, and $k is the number of random tests to perform. The higher the value of $k, the more accurate the result.

Example: Testing a Large Number

The following example script demonstrates how to use the above function:

$n = gmp_init("1234567890987654321"); // The number to be tested
$k = 10; // Run 10 random checks

$result = fermatTest($n, $k);

if ($result){
    echo "Likely a prime number";
} else {
    echo "Not a prime number";
}

In this test, we check whether the large number 1234567890987654321 is a probable prime using Fermat’s test 10 times. If the output is “Likely a prime number,” then the number has a high chance of being prime; otherwise, it's not.

Conclusion and Use Cases

With PHP and the GMP extension, developers can easily implement Fermat-based primality tests to quickly evaluate large integers. This technique is particularly useful in cryptographic applications, such as key generation and large-number filtering.

Beyond Fermat's test, GMP supports a wide range of operations including addition, subtraction, multiplication, division, modulo, and bitwise functions. For PHP developers working with large number computations, GMP is an essential tool.