We consider the standard $K$-armed bandit problem under a distributed trust model of differential privacy (DP), which enables to guarantee privacy without a trustworthy server. Under this trust model, previous work largely focus on achieving privacy using a shuffle protocol, where a batch of users data are randomly permuted before sending to a central server. This protocol achieves ($ε,δ$) or approximate-DP guarantee by sacrificing an additional additive $O\!\left(\!\frac{K\log T\sqrt{\log(1/δ)}}ε\!\right)\!$ cost in $T$-step cumulative regret. In contrast, the optimal privacy cost for achieving a stronger ($ε,0$) or pure-DP guarantee under the widely used central trust model is only $Θ\!\left(\!\frac{K\log T}ε\!\right)\!$, where, however, a trusted server is required. In this work, we aim to obtain a pure-DP guarantee under distributed trust model while sacrificing no more regret than that under central trust model. We achieve this by designing a generic bandit algorithm based on successive arm elimination, where privacy is guaranteed by corrupting rewards with an equivalent discrete Laplace noise ensured by a secure computation protocol. We also show that our algorithm, when instantiated with Skellam noise and the secure protocol, ensures \emph{Rényi differential privacy} -- a stronger notion than approximate DP -- under distributed trust model with a privacy cost of $O\!\left(\!\frac{K\sqrt{\log T}}ε\!\right)\!$.