On the lattice structure of a nonlinear generator with modulus \(2^{\alpha}\)
Let \(Z_ m=\{0,1,...,m-1\}\), \(Z^*_{2^{\alpha}}=\{1,3,...,2^{\alpha}-1\}\) for m,\(\alpha\in N\). Define a nonlinear generator by \(x_{n+1}\equiv ax_ n^{-1}+b(mod 2^{\alpha})\) (1), \(n\geq 0\), where \(x^{-1}\) is the unique element in \(Z^*_{2^{\alpha}}\) with x \(x^{-1}\equiv 1 mod(2^{\alpha})\) for \(x\in Z^*_{2^{\alpha}}\) and \(x_ 0\in Z^*_{2^{\alpha}}\). Assumptions such that (1) has maximal period length \(2^{\alpha -1}\) are given. It is proved that the set of consecutive pseudorandom numbers forms a superposition of shifted lattices. The lattice bases are determined.
scientific article; zbMATH DE number 4151694