A rubber composition based on a reinforcing filler and on an elastomeric matrix comprising a highly saturated diene elastomer, comprising units of the following categories A, B, C, D and E, or units of these same categories A, B, C, D and E comprising, for at least one of the categories B, C, D and E, a mixture of at least two different units of the same category which differ from one another through the nature of their side groups Ri: and where R1, R2, R3 and R4, which are identical or different, denote a hydrogen atom, a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical; R5 denotes a hydrogen atom or a methyl radical; R6 denotes a methyl radical or a phenyl radical which is substituted or unsubstituted in the ortho, meta or para position by a methyl radical; m, n, o, p and q are numbers ranging from 0 to 100. This rubber composition enables a satisfactory degree of stiffness to be achieved while maintaining acceptable hysteresis properties, or even improving them, for use in the manufacture of tires for motor vehicles.