Conversion of Mersenne Twister to double-precision floating-point numbers

摘要

The 32-bit Mersenne Twister generator MT19937 is a widely used random numbergenerator. To generate numbers with more than 32 bits in bit length, andparticularly when converting into 53-bit double-precision floating-pointnumbers in $[0,1)$ in the IEEE 754 format, the typical implementationconcatenates two successive 32-bit integers and divides them by a power of $2$.In this case, the 32-bit MT19937 is optimized in terms of its equidistributionproperties (the so-called dimension of equidistribution with $v$-bit accuracy)under the assumption that one will mainly be using 32-bit output values, andhence the concatenation sometimes degrades the dimension of equidistributioncompared with the simple use of 32-bit outputs. In this paper, we analyze suchphenomena by investigating hidden $\mathbb{F}_2$-linear relations among thebits of high-dimensional outputs. Accordingly, we report that MT19937 with aspecific lag set fails several statistical tests, such as the overlappingcollision test, matrix rank test, and Hamming independence test.