# Minimum of a linear congruence sub-sequence

I have the following little problem : let $a,b,u,v$ be four given integers with $\gcd(a,b)=1$. Now I would like to find the minimum of the linear congruence subsequence $\{ax \pmod b : u \le x \le v\}$. Of course, we have $0<u<v<b$ otherwise it is trivial.

I have some sort of algorithm but wonder if there have already been some work on such a problem ? some references ?

I wonder whether you can make it better than $O(v - u)$. –  Andreas Caranti Mar 2 '13 at 14:07