In this article, we study the problem of distributed estimation of discrete-time LTI systems with bounded noise against sparse integrity attacks. A malicious adversary can corrupt an unknown set with p out of m sensors and manipulate their observations arbitrarily. We propose a general secure estimation framework by decomposing a centralized linear observer into local ones and fusing the local estimates by minimizing specially designed convex functions. The optimization problem can be solved with an linear convergence rate in a distributed manner by widely used proximal gradient descent+consensus iterations aligned with local malicious detectors. Moreover, we do not require to solve the optimization problem exactly. We propose a hot-starting mechanism with state predictions, which combined with linear convergence, can guarantee stable estimation with fixed numbers of iterations at each time instant, both under and without attack. Thus, with bounded computation and communication complexity, the proposed algorithm obtains a secure estimation at each sensor as long as the network is connected and the observability redundancy condition is satisfied, of which the latter is proved to be equivalent to 2p-sparse observability if system matrix A has unitary geometric multiplicity. Furthermore, numerical simulations on IEEE 68-bus system corroborate our proposed algorithm.