# Summary7

```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 xk+n = xk+m ⊕
l
(xku |xk+1
)A, (k = 0, 1, … ). n is the degree of the recurrence. A is a constant w*w
l
matrix with elements {0, 1}. xku means the upper w-r bits of xk, and xk+1
means
the lower r bits of xk+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
2p-1=2nw-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(p2)).
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.
```
Matrix theory

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Linear algebra

58 Cards