TITLE: How to deal with negative exponents in modular arithmetic?
QUESTION [5 upvotes]: So I think I understand how to calculate something like $(208\cdot 2^{-1})\mod 421$ using extended euclidean algorithm. But how would you calculate something like $(208\cdot2^{-21})\mod 421$? 
Thanks, this is basically for my cryptography class; I'm just trying to understand the "big step, baby step" algorithm.

REPLY [4 votes]: Remember that one of the rules of exponents is that
$$(x^a)^b = x^{ab}.$$
So we can rewrite 
$$208 \cdot 2^{-21} \pmod{421}$$ 
as 
$$208 \cdot (2^{-1})^{21} \pmod{421}.$$
You can then solve for the modular multiplcative inverse by one of a few techniques, including, as you note, the Extended Euclidean Algorithm. With this specific example, we get
$$x \equiv 208 \cdot (2^{-1})^{21} \pmod{421}$$
$$\equiv 208 \cdot (211)^{21} \pmod{421}$$
$$\equiv 208 \cdot 329 \pmod{421}$$
$$\equiv 230 \pmod{421}$$