"I know I can loop M times finding a random 0-element to switch to a 1 element"

A potential complexity you didn't realize (or maybe you realized but didn't mention) is that, loop M times, does not gurantee you M different positions, as the random number returned could be duplicates (Of course, you can adjust it, if it is duplicated, but that demages the level of randomization). BrowserUK's solution is preferred.