Step by Step solution about How to resolve the algorithm Modular exponentiation step by step in the C programming language

This C program uses the GNU Multi-Precision (GMP) library to perform modular exponentiation. Here's a breakdown of what the code does:

1. Header and GMP Initialization:

   • The program includes the gmp.h header to access the GMP library.
   • mpz_t is a GMP type used to represent multi-precision integers.

2. Variables:

   • a, b, m, and r are initialized as mpz_t variables to hold multi-precision integers.

3. Setting a and b:

   • mpz_init_set_str initializes a and b with very large integer values (represented as strings). These values are used for the modular exponentiation.

4. Setting m:

   • mpz_init initializes m.
   • mpz_ui_pow_ui calculates 10^40 and assigns it to m. This value is used as the modulus for the modular exponentiation.

5. Calculating r:

   • mpz_powm performs modular exponentiation: r = a^b mod m. It calculates a raised to the power of b modulo m and stores the result in r.

6. Printing r:

   • gmp_printf prints r. The %Zd format specifier is used to print multi-precision integers.

7. Cleanup:

   • After the computation is complete, mpz_clear is used to free the memory allocated for a, b, m, and r.

When you run this program, it outputs the result of the modular exponentiation, which is a very large integer number. This shows how GMP can be used to perform complex mathematical operations on large numbers efficiently.

#include <gmp.h>

int main()
{
	mpz_t a, b, m, r;

	mpz_init_set_str(a,	"2988348162058574136915891421498819466320"
				"163312926952423791023078876139", 0);
	mpz_init_set_str(b,	"2351399303373464486466122544523690094744"
				"975233415544072992656881240319", 0);
	mpz_init(m);
	mpz_ui_pow_ui(m, 10, 40);

	mpz_init(r);
	mpz_powm(r, a, b, m);

	gmp_printf("%Zd\n", r); /* ...16808958343740453059 */

	mpz_clear(a);
	mpz_clear(b);
	mpz_clear(m);
	mpz_clear(r);

	return 0;
}