Monday 5/4/98 7:22 AM chambers, Your statement The advantages are a lack of mathematical structure which might provide an entry for the cryptanalyst, and a huge choice of possibilities; the disadvantages are that there are no guarantees on anything, and as is well known there is a risk of getting a very short period. made at http://www.jya.com/a5-hack.htm#wgc stuck me as profound. Reason is that NSA cryptomathematician Scott Judy once told me that I did not really understand the principles NSA uses for its crypto algorithm. Judy proceeded to explain to me that NSA bases its crypto algorithm on complication, not mathematics. Judy apparently did not realize that some years previous NSA employee Brian Snow showed us about all of NSA's KG schematics. And their field failure records! Masanori Fushimi in Random number generation with the recursion x[t] = x[x-3q]+ x[t-3q],Journal of Applied Mathematics 31 (1990) 105-118 implements a gfsr with period 2^521 - l. http://av.yahoo.com/bin/query?p=gfsr&hc=0&hs=0 Fushimi's generator is sold by Visual Numerics. Fushimi's implementation is very well tested. And worked SO WELL that Visual Numerics numerical analyst Richard Hanson had TO BREAK IT! Reason was that the gfsr produces true zeros. This caused simulation programs to crash from division by zero. None of the linear congruential generators produced zeros so the problem did not arise until the gfsr was used. Hanson ORed in a low-order 1 to fix the problem Masanori wrote, Lewis and Payne [16] introduced an apparely different type of generator, the generalized feed back shift register (GFSR), by which numbers are formed by phase-shifted elements along a M-sequence based on a primitive trinomial 1 + z^q + z^p. Lewis was one of my former ms and phd students. http://www.friction-free-economy.com/ Cycle lengths of sequences is a fascinating topic. Let me point you guys to a delightful article on the distribution of terminal digits of transcendental numbers. The Mountains of pi by Richard Preston, v68 The New Yorker, March 2, 1992 p 36(21). This is a story about Russian-born mathematicians Gregory and David Chudnowsky. While the story is fun to read, I think that the Chudnowsky's were wasting their time. I think that terminal digits of transcendental numbers have been proved to be uniformly distributed. Sobolewski, J. S., and W. H. Payne, Pseudonoise with Arbitrary Amplitude Distribution: Part I: Theory, IEEE Transactions On Computers, 21 (1972): 337-345. Sobolewski, J. S., and W. H. Payne, Pseudonoise with Arbitrary Amplitude Distribution: Park II: Hardware Implementation, IEEE Transactions on Computers, 21 (1972): 346-352. Sobolewski is another of my former phd students. Hopefully you guys will read judge Santiago Campos' 56 page MEMORANDUM OPINION AND ORDER on the Payne and Morales lawsuit on jya.com within several days. I made a copy and gave it to Sobolewski on Sunday afternoon. I want Sobolewski's opinion on what Morales and I should do. Soblewski lives about two miles from us. Sobloweski is an administrator [vp of computing at university of new mexico] and knows how administrators think. Let's hope this UNFORTUNATE mess involving shift register sequences gets settled. But let's not forget our sense of humors despite the about .5 million dead Iranians. Hopefully the system will take care of the guys that did that did the Iranians. Masanori wrote, The GFSR sequence as well as the Tausworthe sequence can be constructed using any M-sequence whether the characteristic polynomial is trinomial or not;... Jim Durham, my seismic data authenticator project leader, retired from Sandia. Durham gave me a number of tech reports upon his retirement. One was authored by Robert TITSWORTHE of jpl. TITSWORTHE changed his name! Later guys