Binary Exponentiation: When to Use It

Start with the idea

A problem is a good match for Binary Exponentiation when it asks for large powers, modular arithmetic, repeated transforms. Here the concrete task is: Return 3^13 modulo 1000. The visible input base=3 exp=13 mod=1000 is an example, not a definition of the whole problem. Ask what operations the solution must support and what the input guarantees. Check number size, sign, and modulus assumptions before using a shortcut; not every inverse or division exists. The right choice follows from those guarantees, not from a familiar noun in the prompt.

The decisive invariant is: Square the base and consume exponent bits so only logarithmically many multiplications occur. Try to say why it applies to the task without quoting the implementation. Which fact becomes known after one update, and which possible answers remain? An arithmetic or bitwise transformation relies on an identity. Write that identity and the domain in which it holds. If that claim cannot be maintained on a legal input, the pattern is wrong or one of its preconditions is missing. A counterexample is more useful than a vague warning: change one property of the sample, trace the first two steps, and identify the exact claim that fails.

Compare costs before committing. The advertised bound is O(log exponent) time · O(1) space; identify the operation responsible for it and the extra memory it needs.