The contact problem of a semi-infinite elastic body with a rigid rough surface with adhesive forces is modeled using a boundary element method (BEM). An original theoretical framework for this problem is presented. Four original BEM algorithms are studied, for both prescribed normal pressure and prescribed penetration, in primal and dual forms. They are all based on a conjugate gradient iterative solver. For each case, the solver is finely tuned to tackle the current problem. Using fast Fourier transforms (FFT) speeds up the computation of the matrix–vector multiplications. A new way of computing the coefficients of the conjugate gradient solver which reduces the number of FFTs at each step is presented. The stability and efficiency of the different methods are compared, showing a great sensitivity to the Tabor coefficient and to the contact area ratio. The most stable algorithm proved to be reliable for very large-scale computations through a participation to M. Mueser’s Contact Mechanics Challenge.