In this paper, we arithmetically describe the convex hull of a digital straight segment by three recurrence relations. This charac- terization gives new insights into the combinatorial structure of digital straight segments of arbitrary length and intercept. It also leads to two on-line algorithms that computes a part of the convex hull of a given digital straight segment. They both run in constant space and constant time per vertex. Due to symmetries, they are enough to reconstruct the whole convex hull. Moreover, these two algorithms provide efficient solutions to the subsegment problem, which consists in computing the minimal parameters of a segment of a digital straight line of known pa- rameters. Due to its close link to the subsegment problem, this work helps to understand the effects of length and intercept on segments of parallel digital straight lines, such as sections of a piece of digital plane or sub-sampled digital straight segments.