We consider the problem of estimating the state of a linear Gaussian system in the presence of integrity attacks. The attacker can compromise p out of m sensors, the set of which is fixed and unknown to the system operator, and manipulate the measurements arbitrarily. Under the assumption that all the eigenvalues of system matrix A have geometric multiplicity 1 (A is non-derogatory), we propose a secure estimation scheme that is resilient to integrity attack as long as the system is 2p-sparse observable. In the absence of attack, the proposed estimation coincides with Kalman estimation with a certain probability that can be adjusted. Furthermore, our proposed estimator is computational efficient during the security condition checking in the designing phase and during the estimation computing in the online operating phase. A numerical example is provided to corroborate the results and illustrate the performance of the proposed estimator.