Summary of Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator Yu Zhang [email protected]

In this paper, the author presented a new random number generator called Mersenne Twister. It is an improved version of Twister GFSR generator. Compared with GFSR and TGFSR, this algorithm has a lot of advantages. The author of this paper analyzed the new method in detail. The MT algorithm is based on the linear recurrence equation x k + n = x k + m ⊕ (x u k |x l k+1 )A, (k = 0, 1, … ).

n is the degree of the recurrence. A is a constant w*w matrix with elements {0, 1}. x u k means the upper w-r bits of x k , and x l k+1 means the lower r bits of x k+1 . Here, r is an integer that not less than w-1 and not more than 0. m is an integer not more than n and not less 1. Here, the author use incomplete array which is one of the two different places from TGFSR algorithm. By using this array, the sequence attains the maximum period 2 p -1=2 nw-r -1. So the sequence is (n-1)-distributed and the seed cannot affect the randomness. Moreover, if we use the (n*w-r)-array, the dimension of the state space is nw-r. We do not need to do factorization. Another point added to this new method is a fast algorithm which can test the primitivity of the linear recurrence’ polynomial: the inversive-decimation-method. By using this method, the primitivity of the characteristic polynomial can be tested with less time consumption (O(p 2 )). At the end of this paper, the author stated the advantages to show the MT method is one of the most efficient methods at present. For example, it has a long period and larger k-distributions. The generation and the parameter search are efficient because of the recurrence form. The randomness is not affected by the seeds. The speed is fast and it can pass several statistic tests.