The NewGenerator command outputs a Maple procedure, a pseudo-random number generator, which when called outputs one pseudo-random integer. The output of the generator depends on the options described below. The default is to output integers on the range $0..999999999999$, i.e., a random 12 digit integer.

The generator is a Quadratic Congruence (QC) generator. A QC generator uses the quadratic recurrence $x_{k+1}=x_k^2\mathbf{mod}n$ which generates a sequence of integers $x_1,...$ where n is a product of two primes p and q and $x_0$ is determined from the seed of the generator. The pseudo-random integers are extracted from the sequence $x_1,...$ by using the least significant half of the digits of the $x_k$'s.

The quality of the pseudo-random numbers generated is expected to be good because the least significant bits of the $x_{k+1}$ are cryptographically secure.  See RandomTools[BlumBlumShub].

The primes p and q are chosen to be of the form $p=\mathrm{2s}+1$ and $q=\mathrm{2t}+1$ where s and t are prime and 2 is a primitive element in the Z mod s and in Z mod t.  The initial value $x_0$ is chosen as a function of the seed so that the period of the generator is maximal where the maximal period is lcm(s-1,t-1) which about n/8. We have precomputed random primes p and q satisfying these requirements of lengths 10, 12, 15 and 16 decimal digits in length. They are

Thus there are four choices of n available which have lengths 20, 24, 30, and 32 digits respectively with periods > 10^19, 10^23, 10^29 and 10^31, respectively. The larger primes will give ``better'' pseudo-random numbers and provide a longer the period. For most applications the smallest choice, the default, will be fine and it will be slightly faster than the larger choices.

The following optional arguments are supported. They are input as equations in any order.

seed=integer

The given integer is the seed of the generator. The value used for $x_0$ is computed from the value of the seed argument It will be a quadratic residue in Z mod n of maximal period and $x_0$ will be larger than the sqrt(n).  The default value for seed is $3$.

range=integer..integer or integer

If the value of the range argument is a range, then the integer will be chosen from that range.  If the value of the range argument is an integer, then the integer will be take from the range [0..value).  The default range is $1000000000000$.

primes=$10$, $12$, $15$ or $16$

The integer l, which must be one of 10, 12, 15 and 16 specifies the choice of the length of the primes p and q in decimal digits.  The default value for primes in $10$.

$\mathrm{with}\left(\mathrm{RandomTools}\left[\mathrm{QuadraticCongruence}\right]\right)$

$\left[\mathrm{NewGenerator}\right]$

$Q≔\mathrm{NewGenerator}\left(\mathrm{range}=1..6\right)$

$Q≔\mathbf{proc}\left(\right)x≔\mathrm{irem}\left(x\^2\,n\right)\;\mathrm{irem}\left(x\,6\right)\+1\mathbf{end\; proc}$

$Q\left(\right)$

$6$

$\mathrm{seq}\left(Q\left(\right)\,i=1..10\right)$

$2,2,5,5,1,3,2,5,4,4$

$Q≔\mathrm{NewGenerator}\left(\mathrm{range}={10}^{10}\right)$

$Q≔\mathbf{proc}\left(\right)x≔\mathrm{irem}\left(x\^2\,n\right)\;\mathrm{irem}\left(x\,10000000000\right)\mathbf{end\; proc}$

$Q\left(\right)$

$949577917$

$\mathrm{seq}\left(Q\left(\right)\,i=1..5\right)$

$1692203819,8074260747,7379622648,939761758,1745728396$

$\mathrm{Float}\left(Q\left(\right)\,-10\right)$

$0.7635452190$

$\mathrm{seq}\left(\mathrm{Float}\left(Q\left(\right)\,-10\right)\,i=1..5\right)$

$0.1932583925,0.2079613478,0.8029354267,0.6006911435,0.6024141496$

$Q≔\mathrm{NewGenerator}\left(\mathrm{primes}=16\,\mathrm{range}={10}^{32}\right)$

$Q≔\mathbf{proc}\left(\right)\mathbf{local}y\&semi;x≔\mathrm{irem}\left(x\&Hat;2\&comma;n\right)\&semi;y≔\mathrm{irem}\left(x\&comma;10000000000000000\right)\&semi;\mathbf{while}y<100000000000000000000000000000000\mathbf{do}x≔\mathrm{irem}\left(x\&Hat;2\&comma;n\right)\&semi;y≔10000000000000000\&ast;y\&plus;\mathrm{irem}\left(x\&comma;10000000000000000\right)\mathbf{end\; do}\&semi;\mathrm{irem}\left(y\&comma;100000000000000000000000000000000\right)\mathbf{end\; proc}$

$Q\left(\right)$

$18259096917880657442169378214465$