Let $A$ and $B$ be two one-dimensional, finite collections of unsigned integers (e.g. arrays). Furthermore, $card(A) = a < b = card(B)$. Both collections are sorted in ascending order. There is at least one item ${x}$ which is contained both in $A$ and $B$. Question: what is the fastest algorithm to find the smallest $x$ and what is its $T(n)$ in Big O notation?    Note: card() means the size of array (say: card(A) = 10 means a declaration in C/C++ would be int[10] A with indices 0...9). Note 2: as I am new to CS.SE and learn CS as enthusiast, so far I have not been exposed to any fancy algorithm to solve this. My initial (naive) guess would be brute-search approach, but this is obviously not efficient for large $A$ and $B$. Your thoughts and pieces of advice would be then highly appreciated. The priority is practical efficiency, so something around polynomial time-algorithm would be nice. Note, that the number of elements in both collections will be very substantial (above 10^30).   
Finding a match in two collections
algorithms;search algorithms;arrays