We consider the problem of resilient state estimation in the presence of integrity attacks. There are m sensors monitoring the state and p of them are under attack. The sensory data collected by the compromised sensors can be manipulated arbitrarily by the attacker. The classical estimators, such as the least squares estimator, may not provide a reliable estimate under the so-called (p, m)-sparse attack. In this paper, we are not restricting our efforts to studying whether any specific estimator is resilient to the attack or not, but instead we aim to present some generic sufficient and necessary conditions for resilience by considering a general class of convex optimization based estimators. The sufficient and necessary conditions are shown to be tight, with a trivial gap. We further specialize our result to the scalar sensor measurements case and extend our framework to incorporate estimators with correlated cost function optimization. Experimental results tested on the IEEE 14-bus test system validate the theoretical analysis.