Step by Step solution about How to resolve the algorithm Modular inverse step by step in the PHP programming language

The provided PHP code defines a function invmod that calculates the modular inverse of a given number a modulo n. The modular inverse is the integer x such that (a * x) % n == 1. It uses the extended Euclidean algorithm to efficiently compute the modular inverse.

Here's a step-by-step explanation of the code:

1. The function invmod takes two parameters:

   • $a: The integer for which we want to find the modular inverse.
   • $n: The modulus.

2. It first checks if $n is negative. If it is, $n is negated (multiplied by -1).

3. It then checks if $a is negative. If it is, $a is replaced with the remainder of -$a divided by $n. This ensures that $a is always positive.

4. It initializes six variables:

   • $t, $nt: These variables represent the coefficients of the form t * a + nt * n that will be used to find the modular inverse.
   • $r, $nr: These variables represent the remainders and coefficients in the Euclidean algorithm.

5. The code enters a while loop that continues as long as $nr is not equal to 0.

6. Inside the loop, it calculates the quotient $quot as the integer division of $r by $nr.

7. It updates $t, $nt, $r, and $nr using the following formulas:

   • $tmp = $nt; $nt = $t - $quot * $nt; $t = $tmp;
   • $tmp = $nr; $nr = $r - $quot * $nr; $r = $tmp;

   These formulas are derived from the extended Euclidean algorithm.

8. The loop continues until $nr becomes 0. If $nr is not 0 after the loop, it means that $a and $n do not have a modular inverse, and the function returns -1.

9. If $r is greater than 1 after the loop, it means that $a and $n are not coprime, and the function returns -1.

10. If $t is negative after the loop, it is adjusted by adding $n to it.

11. Finally, the function returns $t, which is the modular inverse of $a modulo $n.

In the example usage provided in the code, it calculates invmod(42, 2017) and prints the result, which is 1008.

<?php
function invmod($a,$n){
        if ($n < 0) $n = -$n;
        if ($a < 0) $a = $n - (-$a % $n);
	$t = 0; $nt = 1; $r = $n; $nr = $a % $n;
	while ($nr != 0) {
		$quot= intval($r/$nr);
		$tmp = $nt;  $nt = $t - $quot*$nt;  $t = $tmp;
		$tmp = $nr;  $nr = $r - $quot*$nr;  $r = $tmp;
	}
	if ($r > 1) return -1;
	if ($t < 0) $t += $n;
	return $t;
}
	printf("%d\n", invmod(42, 2017));
?>