You had better not have any "secure thoughts" then. "Forever" is enough time to break _any_ cipher.
-- Lefty (lefty@apple.com)
Not to split universes here, but it is interesting to consider that some ciphers may not be breakable in _our_ universe, in any amount of time. Our universe presumably has some finite number of particles (currently estimated to be 10^73 particles). This leads to the "even if every particle were a Cray Y-MP it would take..." sorts of thought experiments. But I am considering _energy_ here. Ignoring reversible computation for the moment, computations dissipate energy (some disagree with this point). There is some uppper limit on how many basic computations could ever be done with the amount of free energy in the universe. (A rough calculation could be done by calculating the energy output of stars, stuff falling into black holes, etc., and then assuming about kT per logical operation. This should be accurate to within a few orders of magnitude.) I haven't done this calculation, and won't today, but the result would likely be something along the lines of X joules of energy that could be harnessed for computation, resulting in Y basic primitive computational steps. I can then find a modulus of 3000 digits or 5000 digits, or whatever, that takes more than this number of steps to factor. Caveats: 1. Maybe there are really shortcuts to factoring. Certainly improvements in factoring methods will continue. (But of course these improvements are not things that convert factoring into a less than exponential-in-length problem...that is, factoring appears to remain "hard.") 2. Maybe reversible computations (a la Landauer, Bennett, et. al.) actually work. Maybe this means a "factoring machine" can be built which takes a fixed, or very slowly growing, amount of energy. In this case, "forever" means Lefty is probably right. 3. Maybe the quantum-mechanical idea of Shore is possible. (I doubt it, for various reasons.) Anyway, this post is of no practical importance. However, I continue to find it useful to think of very large numbers as creating "force fields" or "bobbles" (a la Vinge) around data. A 5000-decimal-digit modulus is as close to being unbreakable as anything we'll see in this universe. --Tim May -- .......................................................................... Timothy C. May | Crypto Anarchy: encryption, digital money, tcmay@netcom.com | anonymous networks, digital pseudonyms, zero 408-688-5409 | knowledge, reputations, information markets, W.A.S.T.E.: Aptos, CA | black markets, collapse of governments. Higher Power: 2^859433 | Public Key: PGP and MailSafe available. "National borders are just speed bumps on the information superhighway."