Exponentiation of power of two by squaring

I am trying to implement exponentiation by squaring of a power of two modulo 2^n − 1. I do not store the power of two, just its exponent (e.g, 4 = 2^2, the exponent is 2).

For example, I want co compute (2^2)^30 (mod 2^5-1). The correct answer is 1. However, my method below gives wrong result which is 16. The GMP code:

/* input:
    mp_bitcnt_t pos ... the exponent of a base, base = 1 << pos, pos is initially 2 (thus 1 << pos is 4)
    const mpz_t exp ... the exponent for squaring (30)
    mp_bitcnt_t m_exp ... Mersenne number exponent (5) (thus the modulus is 2^5-1)
*/

while (mpz_cmp_ui(exp, 0UL) > 0) { /* exp > 0 */
    if (mpz_odd_p(exp)) { /* exp is odd */
        pos += 1;
    }

    pos = 2 * pos;

    mpz_fdiv_q_2exp(exp, exp, 1); /* exp >>= 1 */

    pos = pos % m_exp;
}

/* the result is 1 << pos */

Where is the error in my code?

I can compute this exponentiation using the mpz_powm method and the answer is correct. However, this is EXTREMELY slow. I want to compute this manually.

It turns out that I cannot to modify pos in an odd step, and I have to introduce a separate variable result. The correct code is:

mp_bitcnt_t result = 0;

while (mpz_cmp_ui(exp, 0UL) > 0) {
    if (mpz_odd_p(exp)) {
        result += pos;
        result %= m_exp;
    }

    pos += pos;
    pos %= m_exp;

    mpz_fdiv_q_2exp(exp, exp, 1);
}

/* the result is 1 << result */