pseudorandom numbers

pseudorandom numbers

Generated in a digital computer by a numerical

algorithm

,

pseudorandom numbers

are not random, but should appear to be random when used in

Monte Carlo

calculations.

The most widely used and best understood

pseudorandom generator

is the Lehmer

multiplicative

congruential

generator

, in which each number

r

is calculated as a function of the preceding number in the

sequence

r

i

=

a

⁢

r

i

-

1

(

mod

m

)

or

r

i

=

a

⁢

r

i

-

1

+

c

(

mod

m

)

,

where

a

and

c

are carefully chosen constants, and

m

is usually a

power of two

,

2

k

. All quantities appearing in the formula (except

m

) are

integers

of

k

bits. The expression in brackets is an integer of length

2

⁢

k

bits, and the effect of the modulo

mod

m

is to mask off the most significant part of the result of the

multiplication

.

r

0

is the seed of a generation sequence; many generators allow one to start with a different seed for each run of a

program

, to avoid re-generating the same sequence, or to preserve the seed at the end of one run for the beginning of a subsequent one. Before being used in calculations, the

r

i

are usually transformed to floating point numbers normalized into the range

[

0

,

1

]

. Generators of this type can be found which attain the maximum possible period of

2

k

-

2

, and whose sequences pass all reasonable tests of “randomness”, provided one does not exhaust more than a few percent of the full period.

Multiplicative

random number

generators have serious limitations as random number generators for many tasks, especially those that involve looking at spectra. A number of other fast random number generators exist (such as the Mersenne Twister) all with various proven good qualities.

For many scientific projects, analysis techniques are used that are extremely sensitive to patterns in the input, so when generating fake input using pseudorandom number generators, it is often necessary to use extremely good quality pseudorandom numbers. For some projects it is in fact appropriate to use a hardware device which generates truly random numbers, although even this must be scoured clean of patterns. Some computers now have hardware random number generators built in.

However, there is a completely separate

category

of pseudorandom number generators, namely those used in

cryptography

. Here the goal is to make it computationally infeasible to deduce the next “random” bit in spite of knowing all preceding (and possibly many future) bits. Such pseudorandom number generators can in fact serve as cryptosystems themselves: the requirements for such a random number generator are exactly the same as those for a stream cipher. They are generally more expensive to run but produce much higher quality random numbers. To be used in a cryptographic application, they should also have a sophisticated reseeding schedule and a source of genuine randomness to provide recovery from exposure of their internal state.

References

•

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Title

pseudorandom numbers

Canonical name

PseudorandomNumbers

Date of creation

2013-03-22 12:07:13

Last modified on

2013-03-22 12:07:13

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 11K45

Classification

msc 65C10

Synonym

pseudo-random numbers

Synonym

PRNG

Related topic

TrulyRandomNumbers

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