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RandomTools[QuadraticCongruence]

  

NewGenerator

  

Quadratic Congruence Pseudo Random Number Generator

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

NewGenerator( opt1, opt2, ... )

Parameters

opt1, opt2, ...

-

(optional) argument of the form option=value where option is one of primes, range, or seed

Description

• 

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 xk+1=xk2modn which generates a sequence of integers x1,... where n is a product of two primes p and q and x0 is determined from the seed of the generator. The pseudo-random integers are extracted from the sequence x1,... by using the least significant half of the digits of the xk's.

• 

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

• 

The primes p and q are chosen to be of the form p=2s+1 and q=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 x0 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

            p

             q

      (9999948359)

      (9999854759)

    (999999911447)

    (999999811607)

 (999999999847799)

 (999999999771959)

(9999999999716999)

(9999999999691319)

  

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 x0 is computed from the value of the seed argument It will be a quadratic residue in Z mod n of maximal period and x0 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.

Examples

withRandomTools[QuadraticCongruence]

NewGenerator

(1)

QNewGeneratorrange=1..6

Q:=procx:=iremx^2,n;iremx,6+1end proc

(2)

Q

6

(3)

seqQ,i=1..10

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

(4)

QNewGeneratorrange=1010

Q:=procx:=iremx^2,n;iremx,10000000000end proc

(5)

Q

949577917

(6)

seqQ,i=1..5

1692203819,8074260747,7379622648,939761758,1745728396

(7)

FloatQ,10

0.7635452190

(8)

seqFloatQ,10,i=1..5

0.1932583925,0.2079613478,0.8029354267,0.6006911435,0.6024141496

(9)

QNewGeneratorprimes=16,range=1032

Q:=proclocaly&semi;x:=iremx&Hat;2&comma;n&semi;y:=iremx&comma;10000000000000000&semi;whiley<100000000000000000000000000000000dox:=iremx&Hat;2&comma;n&semi;y:=10000000000000000&ast;y&plus;iremx&comma;10000000000000000end do&semi;iremy&comma;100000000000000000000000000000000end proc

(10)

Q

18259096917880657442169378214465

(11)

See Also

rand

RandomTools

RandomTools[Generate]

RandomTools[QuadraticCongruence]

 


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