Unformatted text preview: ons are in [969, 967, 968, 598].) Despite all of this prior art, a group of researchers from New Zealand managed to patent this scheme in 1993, calling it LUC [1486, 521,1487]. The nth Lucas number, Vn(P,1), is defined as Vn(P, 1) = PVn-1 (P, 1) - Vn- 2 (P,1) There’s a lot more theory to Lucas numbers; I’m ignoring all of it. A good theoretical treatment of Lucas sequences is in [1307, 1308]. A particularly nice description of the mathematics of LUC is in [1494, 708]. In any case, to generate a public-key/private-key key pair, first choose two large primes, p and q. Calculate n, the product of p and q. The encryption key, e, is a random number that is relatively prime to p - 1, q - 1, p + 1, and q + 1. There are four possible decryption keys, d = e-1 mod (lcm((p + 1), (q + 1))) d = e-1 mod (lcm((p + 1), (q - 1))) d = e-1 mod (lcm((p - 1), (q + 1))) d = e-1 mod (lcm((p - 1), (q - 1))) where lcm is the least common multiple. The public key is d and n; the private key is e and n. Discard p and q. To encrypt a message, P (P must be less than n), calculate C = Ve(P, 1) (mod n ) And to decrypt: P = Vd P, 1) (mod n), with the proper d At best, LUC is no more secure than RSA. And recent, still-unpublished results show how to break LUC in at least some implementations. I just don’t trust it. 19.10 Finite Automaton Public-Key Cryptosystems
Chinese cryptographer Tao Renji has developed a public-key algorithm based on finite automata [1301, 1302, 1303, 1300, 1304, 666]. Just as it is hard to factor the product of two large primes, it is also hard to factor the composition of two finite automata. This is especially so if one or both of them is nonlinear. Much of this research took place in China in the 1980s and was published in Chinese. Renji is starting to write in English. His main result was that certain nonlinear automata (the quasilinear automata) possess weak inverses if, and only if, they have a certain echelon matrix structure. This property disappears if they are composed with another automaton (even a linear one). In the public-key algorithm, the secre...
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