An Explicit Bound On Double Exponential Sums Related to Diffie–Hellman Distributions
Mei-Chu Chang, Chui Zhi Yao · SIAM Journal on Discrete Mathematics · 2008
Let p be a prime and V an integer of order t in the multiplicative group modulo p. In this paper, we give an explicit bound on the double exponential sums. For $\varepsilon>0$, we have $\tilde{S}_{a,b,c}(t)=\sum^t_{x,y=1}e_{p}(aV^{x}+bV^{y}+cV^{xy})\ll t^{2-r(\varepsilon)}$, where $r(\varepsilon) = \bigg\{ \begin{array}[c]{c} \frac{1}{4440}\text{ \qquad \qquad \qquad \qquad if }p^{\frac{1}{2} }\leq t\leq p^{\frac{2}{3}},\ \frac{1}{20}\varepsilon^{3}(\log\frac{1}{\varepsilon}) ^{-6}\text{ \qquad \qquad if }p^{\frac{1}{4}+\varepsilon}\leq t