This'll have to be my last reply to Michael Wilson. No offense meant, but we are not even close to speaking the same language.
*** You assume that your selection of primes is random; it is the case, particularly in the initial usages of public-key systems, that attacks could be made on keyspaces based on the prime generation method. A point that number-crunch jockeys tend to forget is that psychology and systems analysis provide greater in-roads against secure systems than brute force.
Your phrasing is Greek to me. The primes are generated by picking a very large random number, of 150 digits or so (depends on key length chosen), and then iterating-and-testing until a prime is found. (I wrote a version of this for my own crude version of RSA, in Mathematica...not very fast, but immensely educational for me.) So I run this and start with a random number of: 3865018936355867.....38587493661988826448627 (152 digits) I run this process a second time and get: 193648376263874....8747487458364253 (152 digits) And I could keep running this as many times as I like, with the numbers being different every time. (These are just examples, not real numbers.) Now tell me, even granted that my RNG is not "perfect" (in the sense we talk about so often here), how could an attacker--even one using the "psychology and systems analysis" Wilson cites--know where to start? Which number I generated? The search space is just too large. Just too much entropy. PGP, for example, asks for keyboard input to get enough entropy. (I assume some of the collected entropy goes directly into the prime generation process, of course.) Even all the world's supercomputers are not going to be able guess (in any number of trials in a million years) the specific 140- or 150- or 160-digit number I generated. (Caveat: Unless the RNG is a brain-dead seeded generator. But that's why MailSafe, PGP, and other programs ask for keyboard input as a source of entropy. Even if the distillation of entropy results in "only" 250 bits of entropy, it's still hopeless to try to enumerate the primes.) I agree with Graham Toal: it's time Michael Wilson either _tells us_ what his magical schemes are, or shuts up. Pompous language is no substitute for meaningful information.
"Parallel versus scalar processing"? Parallelism means nothing at these scales...see the above point.
*** Your point is orthogonal to our point. The two systems are used for different attacks--parallelism can be used for exhaustive search, such as for DES keys, while scalar processing can be used for testing primality.
Gobbledegook! A "parallel" machine with 1024 nodes is at most 1024 times faster than a single node...no magical gains. The RSA-129 challenge did use lots (hundreds, maybe thousands) of nodes, but this was--as expected--a proportionate gain. Saying an intractable problem becomes tractable with "parallel processing" is simply wrong. I suppose one could magically hypothesize a machine with "10^100 nodes" and say "See, parallel processing allowed us to factor this and such number," but this is pure fantasy. Exponential blowup (non-polynomial time) means just that...a few factors of 16 or 4096 or whatever just don't make a difference. Please provide us with specifics of your methods. If you say they are "proprietary" or that you are seeking a patent on them, I won't be surprised. --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."