Last 3 digits

Note that $\displaystyle 256 = 2^8$ . We can then square $17$ eight times, finding the result modulo $1000$ every time until we find $\displaystyle 17^{256} \pmod{1000}$ .

$\displaystyle 17^2 \ \equiv \ 289 \pmod {1000}$

$\displaystyle 17^4 \ \equiv \ (17^2)^2 \ \equiv \ 289^2 \ \equiv \ 83521 \ \equiv \ 521 \pmod{1000}$

$\displaystyle 17^8 \ \equiv \ (17^4)^2 \ \equiv \ 521^2 \ \equiv \ 271441 \ \equiv \ 441 \pmod{1000}$

$\displaystyle 17^{16} \ \equiv \ (17^8)^2 \ \equiv \ 441^2 \ \equiv \ 481 \pmod{1000}$

$\displaystyle 17^{32} \ \equiv \ 481^2 \ \equiv \ 361 \pmod{1000}$

$\displaystyle 17^{64} \ \equiv \ 361^2 \ \equiv \ 321 \pmod{1000}$

$\displaystyle 17^{128} \ \equiv \ 321^2 \ \equiv \ 41 \pmod{1000}$

$\displaystyle 17^{256} \ \equiv \ 41^2 \ \equiv \ \boxed{681} \pmod{1000}$