# Count-the-1's Test (Stream of Bits)

## Test Purpose

The test evaluates the randomness of the overlapping random five-letter words sequence. The five-letter words have the specified distribution of the probabilities of obtaining the specified letter. The test forms the random letters from the integer output of a BRNG. The test treats the integer output as a sequence of bits.

## First Level Test

The first level test assumes that the integer output is a sequence of random bits. The test interprets this bit sequence as a sequence of bytes, that is, a sequence of 8-bit integer numbers. The number of 1’s in every random byte should have a binominal distribution with m = 8, p = 1/2 parameters. Therefore, the probability of getting k 1’s in a byte is equal to 2-8C8k. The first level test regards a random variable c that takes five possible values:

c = 0, if the number of 1’s in a random byte is less than three,

c = 1, if the number of 1’s in a random byte is three,

c = 2, if the number of 1’s in a random byte is four,

c = 3, if the number of 1’s in a random byte is five,

c = 4, if the number of 1’s in a random byte is more than five.

The probability distribution of c is the following:

The test interprets c as a selection of a random letter from the alphabet {a, b, c, d, e} with the probabilities q0, q1, q3, q4 respectively. Thus, the sequence of random bytes b0, b1, b2, ... corresponds with the defined sequence of random letters l0, l1, l2, ... . The test forms overlapping words of length four: v1= l1l2l3l4, v2= l2l3l4l5, ... and length five: w1= l1l2l3l4l5, w2= l2l3l4l5l6, ... from this sequence. The test computes the frequencies of getting each of 625 of possible four-letter words and of 3,125 of possible five-letter words for 2,560,000 of the obtained words. According to these frequencies, the test makes the chi-square statistics V1 and V2 for the four- and five-letter words respectively. The test takes into account the covariance of the frequencies of the fallouts of four-letter and five-letter words and performs the chi-square test for the V2 - V1 statistic. The V2 -V1 statistic is asymptotically normal with a mean a = 2500 and standard deviation s = 70.71. The result of the first level test is the p-value.

In the table below, NB stands for the number of bits required to represent a random number in integer arithmetic, WS stands for the machine word size, in bits, used in integer random number generation.

BRNG

Integer Output Interpretation

MCG31m1

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-30. NB=31, WS=32.

R250

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

MRG32k3a

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

MCG59

Array of 64-bit integers. Each 64-bit integer uses the following bits:

0-58. NB=59, WS=64.

WH

Array of quadruples of 32-bit integers. Each 32-bit integer uses the following bits:

0-23. NB=24, WS=32.

MT19937

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

MT2203

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

SFMT19937

Array of quadruples of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

PHILOX4X32X10

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

ARS5

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

The test selects only NB of lower bits from each WS-bit integer to form a bit sequence.

## Second Level Test

The second level test performs the first level test ten times. The test applies the Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling statistics to the obtained p-values of pj, j = 1, 2, ..., 10. If the resulting p-value is p < 0.05 or p > 0.95, the test fails.

## Final Result Interpretation

The second level test performs ten times. The test computes the FAIL percentage of the failed second level tests. The acceptable result is the value of FAIL < 50%.

## Tested Generators

Function Name

Application

vsRngUniform

not applicable

vdRngUniform

not applicable

viRngUniform

not applicable

viRngUniformBits

applicable

Function Name

Application

vsRngUniform

not applicable

vdRngUniform

not applicable

viRngUniform

not applicable

viRngUniformBits

applicable

The WH and SFMT19937 generators use all the four elements to form a bit sequence.

For more complete information about compiler optimizations, see our Optimization Notice.