Double exponential sums over thin sets

We estimate double exponential sums of the form Sa(x,y)= ∑x∈x ∑x∈y exp (2πia^xy/p) where is of multiplicative order t modulo the prime p and χ and y are arbitrary subsets of the residue ring modulo t. In the special case t = p-1, our bound is nontrivial for |χ| ≥ |y| ≥ p^15/16+δ with any fixed δ > 0, while if in addition we have |χ| ≥ p^1-δ/4 it is nontrivial for |y| ≥ p^3/4+δ.