This paper presents a unified formal framework for integrated circuits (ICs) Trojan detection that can simultaneously employ multiple noninvasive side-channel measurement types (modalities). After formally defining the IC Trojan detection for each side-channel measurement and analyzing the complexity, we devise a new submodular formulation of the problem objective function. Based on the objective function properties, an efficient Trojan detection method with strong approximation and optimality guarantees is introduced. Signal processing methods for calibrating the impact of interchip and intrachip correlations are presented. We define a new sensitivity metric that formally quantifies the impact of modifications to each existing gate that is affected by Trojan. Using the new metric, we compare the Trojan detection capability of different measurement types for static (quiescent) current, dynamic (transient) current, and timing (delay) side-channel measurements. We propose four methods for combining the detection results that are gained from different measurement modalities and show how the sensitivity results can be used for a systematic combining of the detection results. Experimental evaluations on benchmark designs reveal the low-overhead and effectiveness of the new Trojan detection framework and provides a comparison of different detection combining methods.