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On Overlappings of Digitized Straight Lines and Shared Steganographic File Systems

We consider the unbounded integer grid and the digitized version of the straight line y=α x+β, with α,β ∈ ℜ being the set of points (i,[α i+β]), i∈ Ζ, where [·] is the integer rounding operator ([x]-0.5≤ x < [x]+0.5). We address the problem of counting the number of points in the integer grid in which two digitized straight lines overlap each other in the particular case when the crossing point of the non-digitized version of the lines has integer coordinates and the slopes belong to the set {a/b:a∈ {-(N-1),..,(N-1)}, b∈ {1,..,(N-1)}}, that is, all the possible slopes of the segments between two different points in the N × N grid. 
 
Applications of this problem are explained, with a special focus on a shared steganographic system with error correction.