This occurred to me this morning in the inner sanctum of inspired cogitation, aka the shower. "What is the maximum # of bit a public-key would ever need to be, given no breakthroughs in factoring?" I came up with an answer, but it depends on some numbers that I don't have handy; perhaps other people on the list can fill in the blanks. First, we need an equation that tell us how difficult it is, in # of operations, to factor a number of N bits. eg: N_ops(N) = # of operations it will take. Then all we need to do is find the N for which N_ops(N) is greater than U_Duration * U_Particles * (1 / P_time) Where U_Duration is the expected duration of the universe, U_Particles is the number of particles in the universe (I am assuming that every particle can be used as a processor; the programming I leave as an exercise to the alert reader), and P_Time is the Planck time (damned if I can remember it) in seconds, which ought to be a good upper bound for clock speed on the Universal CPU. A most likely useless number, but it would be interesting to know what it comes out to. Best, R