Double exponential sums related to Diffie-Hellman distributions

We establish the bound $\sum\nolimits_{x\in X}{\vert\sum\nolimits_{y\in Y}\gamma{(y)}\exp\operatorname\pi{(2\pi ia\lambda^{xy}/p)}\vert}\ll{\vert X\vert}^{1/2}$ $\times{\vert Y\vert}^{1/2}p^{9/8+\varepsilon}/T^{1/4}$ , where λ is of multiplicative order T modulo a large prime p, X and Y are arbitrary subsets of the residue ring modulo p−1 with |X| and |Y| elements correspondingly, γ(n) are any complex numbers with |γ(n)|≤ 1, and ε is any small positive constant. In particular cases, our estimate improves a series of the existing results.