MOD calculation / reversing

[bilbo]

@raduga_fb
this sounds interesting... furthermore, in that case the solution is unique!
I need more investigations

@wilson bibe
sorry to contradict you but these are the facts, under a programmer approach:

in "C" (max bits used in calculations 7FFFFFFF):

#include <stdio.h>

void
main(void)
{
	int a, mod=67890, c=48765;

	for (a=0; a<1295; a++)
		if ((a*a*a) % mod != (a*(a*a % mod)) % mod) printf("KO for a=%d\n", a);
}

output is:
KO for a=1291
KO for a=1292
KO for a=1293
KO for a=1294

This means no difference with a < 1291

using GMP big-integers library:

void
main(void)
{
	mpz_t base, mod;
	mpz_t res1, res2, res3, res4;

	mpz_init(base);
	mpz_init(mod);

	mpz_init(res1);
	mpz_init(res2);
	mpz_init(res3);
	mpz_init(res4);

	mpz_set_ui(mod, 67890);

	for ( ; mpz_cmp_ui(base, 0xFFFFF)<=0; mpz_add_ui(base, base, 1)) {
    	mpz_powm_ui(res1, base, 3, mod);  // (a*a*a) % mod
    	mpz_powm_ui(res2, base, 2, mod);  // (a*a) % mod
    	mpz_mul(res3, base, res2);  // a * ((a*a) % mod)
    	mpz_mod(res4, res3, mod);  // (a * ((a*a) MOD mod)) % mod
    	if (mpz_cmp(res4, res1))
        	gmp_printf("KO for a=%Zd\n", base);
    	}
}

Nothing is printed. This means that there is no difference at all between
(a*a*a) % mod and (a * ((a*a) % mod)) % mod (at least for base up to 0x100000)

Best regards
bilbo