[PDF] Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator | Semantic Scholar

An asymptotically random 23-bit number sequence of astronomic period, 2607 - 1, is presented and possesses equidistribution and multidimensional uniformity properties vastly in excess of anything that has yet been shown for conventional congruentially generated sequences.

A slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit and is most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size.

The author develops a theory of the lattice structure of pseudorandom sequences from shift register generators, i.e. Tausworthe sequences and GFSR (generalized feedback shift register) sequences, and derives a theorem that links the k-distribution of such sequences and the successive minima of thek-dimensional lattice over GF(2,x) associated with the sequences, thereby leading to the geometric interpretation of the crust structure of these sequences.

This follow up article introduces and analyzes a new TGFSR variant having better k-distribution property, and provides an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order.