A multiple recursive non-linear congruential pseudo random number generator

On-linear multiple recursive congruential pseudo random number generator with prime modulus p is introduced. Let x, n0, be the sequence generated by a usual linear (r+1)-step recursive congruential generator with prime modulus p and denote by N(n), n0, the sequence of non-negative integers with x_N(n)0 (mod p). The non-linear generator is defined by z_nx_N(n)+1·x _N(n) ^–1 (mod p), n0, where x _N(n) ^–1 denotes the inverse element of x_N(n) in the Galois field GF(p). A condition is given which ensures that the generated sequence is purely periodic with period length p^r and all (p–1)^r r-tupels (y₁,...,y_r) with 1y₁,...,y_rp are generated once per period when r-tupels of consecutive numbers of the generated sequence are formed. For r=1 this generator coincides with the generator introduced by Eichenauer and Lehn 2].