In this paper, we study the ability of a curvature sensor (CS) to measure co-phasing errors in a segmented primary mirror. We study a new fast and low memory-consuming algorithm based on the definition of an integrated curvature signal. This signal results from the integration of the CS response in enclosures matching the shapes of the segments in the mirror. This gives a low number of measurements, equal to the number of segments. The determination of the position of the segments is speeded with an analytical response model, which we formulate in matrix form. In this way, we can make an analysis of the error propagation that is valid for measurement errors or imprecision of the response model. The procedure is well suited to mirrors with a large number of segments, as in the case of ELTs.