Summary: Drag reduction (DR) under a turbulent boundary layer implies the suppression of turbulent momentum flux to the wall, a large-eddy phenomenon. Our hypothesis is that the essential mechanisms by which dilute concentrations of long-chain polymer molecules reduce momentum flux involve only the interactions among turbulent velocity fluctuations, polymer molecules and mean shear. Experiments indicate that these interactions dominate in a polymer-active ’elastic layer’ outside the viscous sublayer and below a Newtonian inertial layer in a polymer-laden turbulent boundary layer. We investigate our hypothesis by modelling the suppression of momentum flux with direct numerical simulation (DNS) of homogeneous turbulent shear flow (HTSF) and the finite extensible nonlinear elastic with Peterlin approximation (FENE-P) model for polymer stress. The polymer conformation tensor equation was solved using a new hyperbolic algorithm with no artificial diffusion. We report here on the equilibrium state with fixed mean shear rate \(S\), where progressive increases in non-dimensional polymer relaxation time \(We_S\) (shear Weissenberg number) or concentration parameter \(1 - \beta\) produced progressive reductions in Reynolds shear stress, turbulence kinetic energy and turbulence dissipation rate, concurrent with increasing polymer stress and elastic potential energy. The changes in statistical variables underlying polymer DR with \(1 - \beta, We-S\), %DR and polymer-induced changes to spectra are similar to experiments in channel and pipe flows and show that the experimentally measured increase in normalized streamwise velocity variance is an indirect consequence of DR that is true only at lower DR. Comparison of polymer stretch and elastic potential energy budgets with channel flow DNS showed qualitative correspondence when distance from the wall was correlated to \(We_S\). As \(We_S\) increased, the homogeneous shear flow displayed low-DR, high-DR and maximum-DR (MDR) regimes, similar to experiments, with each regime displaying distinctly different polymer-turbulence physics. The suppression of turbulent momentum flux arises from the suppression of vertical velocity fluctuations primarily by polymer-induced suppression of slow pressure-strain rate correlations. In the high-Weissenberg-number MDR-like limit, the polymer nearly completely blocks Newtonian inter-component energy transfer to vertical velocity fluctuations and turbulence is maintained by the polymer contribution to pressure-strain rate. Our analysis from HTSF with the FENE-P representation of polymer stress and its comparisons with experimental and DNS studies of wall-bounded polymer-turbulence supports our central hypothesis that the essential mechanisms underlying polymer DR lie directly in the suppression of momentum flux by polymer-turbulence interactions in the presence of mean shear and indirectly in the presence of the wall as the shear-generating mechanism.