Character sums with exponential functions over smooth numbers

We give nontrivial bounds in various ranges for exponential sums of the formunder(∑, n ∈ S (x, y)) exp (2 π i a θ{symbol}ⁿ / m) and under(∑, n ∈ S_t (x, y)) exp (2 π i a θ{symbol}ⁿ / m)where m ≥ 2, φ is an element of order t in the multiplicative group Z*_m, gcd(a,m) = 1, S(x,y) is the set of y-smooth integers n≤x, and S_t(x,y) is the subset of S(x,y) consisting of integers that are coprime to t. We obtain sharper bounds in the special case that m = q is a prime number.