Difference between revisions of "Pseudorandom number generators"

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Multiplicative Congruent Generator (or MCG) is simplyfied version of LCG since if ''c'' = 0 in LCG we get the MCG<ref name="Knuth-1997"/>. This generator is defined as follows:
 
Multiplicative Congruent Generator (or MCG) is simplyfied version of LCG since if ''c'' = 0 in LCG we get the MCG<ref name="Knuth-1997"/>. This generator is defined as follows:
  
−
<math>X_{n+1} \equiv \left( a X_n \right)~~\pmod{m}</math>
+
<math>X_{n+1} \equiv a X_n ~~\pmod{m}</math>
  
 
where:
 
where:

Revision as of 20:13, 2 December 2012

In cryptography, a pseudorandom generator (or PSG) is procedure that outputs a sequence computationally indistinguishable from truly random sequence with uniformly distributed random sequence. The prefix pseudo (from Greek ψευδής "lying, false") is used to mark something as false, fraudulent, or pretending to be something it is not. Pseudo random generators find application in many fields besides cryptography such as applied mathematics, physics and simulations. Simulations often require mechanisms producing sequences of random values. These procedures are certainly non-trivial and often require significant amounts of computational time.


Definition

A pseudorandom generator (or PRG) is a (deterministic) map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{G:\{0, 1\}^l \rightarrow \{0, 1\}^n}}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{n \geq l}}} . Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{l}}} is the 'seed length' and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{n - l \geq 0}}} is the 'stretch'. We typically think that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{n \gg l}}} and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{G}}} is effciently computable in some model. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f:\{0, 1\}^n \rightarrow \{0, 1\}}}} is any 'statistical test', we say that G 'Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\epsilon}}} -fools' Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f}}} is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{|Pr[f({U_n} = 1)] - Pr[f(G({U_l})) = 1)]| \leq \epsilon}}}


where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {U_m}} denotes a uniformly random string in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\{0, 1\}^m}}} . Here the string Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{U_l}}} is called the 'seed'. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{C}}} is a class of tests, we say that G 'Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\epsilon}}} -fools Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{C}}} ' or is an 'Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\epsilon}}} -PRG against Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{C}}} ' if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{G}}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\epsilon}}} -fools Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f}}} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f \in C}}} .[1]


In other words we are thying to convince any outside party (let's call them an adversary) that sequences returned from PRG are being produced chosen at random. Adversary may use statistical test algorithms to check simultaneously outputs from PRG and uniformly random sequnces. PRGs ensure that to the adversary both outputs look the same.

Required properties

Reliable PRG should have all these properties[2]:

  • Unbiased - Uniform distribution
    • All values of whatever samplesize is collected are equiprobable
  • Unpredictable - Indipendence
    • It is impossible to predict what the next output will be, given all the previous outputs, but not the internal state
  • Unreproducible
    • Two of the same generators, given the same starting conditions, will produce different outputs
  • Long period
    • The generator should be of long period
  • Fast computation
    • The generator should be reasonably fast
  • Security
    • The generator should be secured

Construction of simple PRG

There are many ways how to construct PRGs and one of the simpliest ones is to use pseudorandom functions and expand the key. A pseudorandom function (or PRF) is any function defined over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal(K, X, Y)}}}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F:K \times X \rightarrow Y}}}

where:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{K}}} is key space
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{X}}} is input space
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{Y}}} is output space

such that exists efficient algorithm to evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal F(k,x)}}} .[3]


On the other hand pseudorandom permutation is any function defined over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal(K, X)}}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{E:K \times X \rightarrow X}}}

such that[3]:

  1. Exists efficient deterministic algorithm to evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal E(k,x)}}}
  2. The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal F(k,\cdot)}}} is one-to-one
  3. Exists efficient inversion algorithm Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\textnormal D(k,y)}}}


Any pseudorandom permutation (or PRP) is also pseudorandom function given

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{X=Y}}}
  2. Function is efficiently invertible


So let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F: K \times \{0, 1\}^n \rightarrow \{0, 1\}^n}}} be a PRF


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{ \begin{cases} Functions[X, Y]: \text{ all functions from X to Y}\\ S_F = {F(k, \cdot) st. k \in K} \subseteq Functions[X, Y] \end{cases} }


For PRF to be suitable for use in PRG it must be secure and therefore computationally indistinguishable from random function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f(\cdot)}}} . This situation is depicted below:


Secure prf.png


The adversary can not distinguish whether output came from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{S_F}}} or some random Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f(\cdot)}}} .


If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F}}} is secure PRF we can use key expantion to construct secure PRG defined as


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{G: K \times \{0, 1\}^{nt}}}} where:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{n}}} number of bits in each block
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{t}}} number of generated blocks


We get return value by using key expansion. Great advantage of using this approach is ability to employ multiple CPU cores and take advantage of paralelization (for example odd values are computed by core 1; even values are computed by core 2). Security of PRG is provided by fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{F(k, \cdot)}}} is indistinguishable form random Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{f(\cdot)}}} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{G(k) = F(k, 0) || F(k, 1) || ... || F(k, t-1)}}}

Linear methods

Linear Congruent Generator

Linear Congruent Generator (or LCG) is one of the best known PRGs in the world. This generator is defined as follows

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} \equiv \left( a X_n + c \right)~~\pmod{m}}

where:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n}} is the sequence of pseudorandom values
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_0,\,0 \le X_0 < m} is the seed
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a,\,0 < a < m} is the multiplier
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c,\,0 \le c < m} is the increment
  5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m,\, 0<m } is the modulo

are integer constants that specify the generator.[4]

The range of output values is restricted since after at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m} values the period starts to repeat itself (in the terms of repeating the same pattern). The most significant element in terms of the length of the period is the multiplier. Best outputs from generator in regard of the length of the period are provided given values:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a - 1} is divisable by all the primes that divide the multiplier
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a - 1} is multiple of number 4, when the multiplier is multiple of number 4
  • the multiplier and the increment do not have common divisor (except from 1)

An example of output exported from WolframAplha application[5]:

LCG example.png

Multiplicative Congruent Generator

Multiplicative Congruent Generator (or MCG) is simplyfied version of LCG since if c = 0 in LCG we get the MCG[4]. This generator is defined as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n+1} \equiv a X_n ~~\pmod{m}}

where:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_{n}} is the sequence of pseudorandom values
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X_0,\,0 \le X_0 < m} is the seed
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a,\,0 < a < m} is the multiplier
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m,\, 0<m } is the modulo

An example of output (of 59-bit multiplicative congruential generator) exported from WolframAplha application[6]:

MCG59 example.png

Lagged Fibonacci generator

Mersenne twister

<TODO> Popis algoritmu.

MT_example.png - Mersenne twister shift register generator An example of output exported from WolframAplha application[6]:

MT example.png

Nonlinear methods

Blum Blum Shub

<TODO> Popis algoritmu.

An example of output constructed in Maple 14 (seed=10, range=1000):

BBS example.gif

Testing

References

  1. ↑ Lecture 16: Nisan's PRG for small space, 15-855: Intensive Intro to Complexity Theory. Spring 2009, Carnegie Mellon University, USA.: page 1
  2. ↑ Xiang-Yang Li, Pseudo-random Number - Cryptography and Network Security. Illinois Institute of Technology, USA
  3. ↑ 3.0 3.1 Boneh, D. (2012); Lecture 3 - Block ciphers, Introduction to Cryptography. Stanford, USA.
  4. ↑ 4.0 4.1 Knuth 1997, Sec. 3.2.1
  5. ↑ WolframAlpha - Linear Congruential Generators (2012)
  6. ↑ 6.0 6.1 WolframAlpha - Mersenne Twister and Friends (2012)

Extarnal links

  1. WolframAlpha - Linear Congruential Generator
  2. WolframAlpha - Mersenne Twister simulator
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