org.apache.commons.math3.random

## Class RandomDataGenerator

- java.lang.Object
- - org.apache.commons.math3.random.RandomDataGenerator

All Implemented Interfaces:
  Serializable, RandomData

    public class RandomDataGenerator
    extends Object
    implements RandomData, Serializable

Implements the `RandomData` interface using a `RandomGenerator` instance to generate non-secure data and a `SecureRandom` instance to provide data for the `nextSecureXxx` methods. If no `RandomGenerator` is provided in the constructor, the default is to use a `Well19937c` generator. To plug in a different implementation, either implement `RandomGenerator` directly or extend `AbstractRandomGenerator`.

Supports reseeding the underlying pseudo-random number generator (PRNG). The `SecurityProvider` and `Algorithm` used by the `SecureRandom` instance can also be reset.

For details on the default PRNGs, see `Random` and `SecureRandom`.

Usage Notes:

- Instance variables are used to maintain `RandomGenerator` and `SecureRandom` instances used in data generation. Therefore, to generate a random sequence of values or strings, you should use just one `RandomDataImpl` instance repeatedly.
- The "secure" methods are *much* slower. These should be used only when a cryptographically secure random sequence is required. A secure random sequence is a sequence of pseudo-random values which, in addition to being well-dispersed (so no subsequence of values is an any more likely than other subsequence of the the same length), also has the additional property that knowledge of values generated up to any point in the sequence does not make it any easier to predict subsequent values.
- When a new `RandomDataImpl` is created, the underlying random number generators are not initialized. If you do not explicitly seed the default non-secure generator, it is seeded with the current time in milliseconds plus the system identity hash code on first use. The same holds for the secure generator. If you provide a `RandomGenerator` to the constructor, however, this generator is not reseeded by the constructor nor is it reseeded on first use.
- The `reSeed` and `reSeedSecure` methods delegate to the corresponding methods on the underlying `RandomGenerator` and `SecureRandom` instances. Therefore, `reSeed(long)` fully resets the initial state of the non-secure random number generator (so that reseeding with a specific value always results in the same subsequent random sequence); whereas reSeedSecure(long) does not reinitialize the secure random number generator (so secure sequences started with calls to reseedSecure(long) won't be identical).
- This implementation is not synchronized. The underlying `RandomGenerator` or `SecureRandom` instances are not protected by synchronization and are not guaranteed to be thread-safe. Therefore, if an instance of this class is concurrently utilized by multiple threads, it is the responsibility of client code to synchronize access to seeding and data generation methods.

Since:
  3.1
Version:
  $Id: RandomDataGenerator.java 1422313 2012-12-15 18:53:41Z psteitz $
See Also:
  Serialized Form

### Field Summary

Fields
`private RandomGenerator` `rand` underlying random number generator
`private SecureRandom` `secRand` underlying secure random number generator
`private static long` `serialVersionUID` Serializable version identifier

### Constructor Summary

Constructors
Constructor and Description
`RandomDataGenerator()` Construct a RandomDataGenerator, using a default random generator as the source of randomness.
`RandomDataGenerator(RandomGenerator rand)` Construct a RandomDataGenerator using the supplied `RandomGenerator` as the source of (non-secure) random data.

### Method Summary

Methods
`private int[]` `getNatural(int n)` Returns an array representing n.
`private RandomGenerator` `getRan()` Returns the RandomGenerator used to generate non-secure random data.
`private SecureRandom` `getSecRan()` Returns the SecureRandom used to generate secure random data.
`private void` `initRan()` Sets the default generator to a `Well19937c` generator seeded with `System.currentTimeMillis() + System.identityHashCode(this))`.
`double` `nextBeta(double alpha, double beta)` Generates a random value from the `Beta Distribution`.
`int` `nextBinomial(int numberOfTrials, double probabilityOfSuccess)` Generates a random value from the `Binomial Distribution`.
`double` `nextCauchy(double median, double scale)` Generates a random value from the `Cauchy Distribution`.
`double` `nextChiSquare(double df)` Generates a random value from the `ChiSquare Distribution`.
`double` `nextExponential(double mean)` Generates a random value from the exponential distribution with specified mean.
`double` `nextF(double numeratorDf, double denominatorDf)` Generates a random value from the `F Distribution`.
`double` `nextGamma(double shape, double scale)` Generates a random value from the `Gamma Distribution`.
`double` `nextGaussian(double mu, double sigma)` Generates a random value from the Normal (or Gaussian) distribution with specified mean and standard deviation.
`String` `nextHexString(int len)` Generates a random string of hex characters of length `len`.
`int` `nextHypergeometric(int populationSize, int numberOfSuccesses, int sampleSize)` Generates a random value from the `Hypergeometric Distribution`.
`int` `nextInt(int lower, int upper)` Generates a uniformly distributed random integer between `lower` and `upper` (endpoints included).
`long` `nextLong(long lower, long upper)` Generates a uniformly distributed random long integer between `lower` and `upper` (endpoints included).
`int` `nextPascal(int r, double p)` Generates a random value from the `Pascal Distribution`.
`int[]` `nextPermutation(int n, int k)` Generates an integer array of length `k` whose entries are selected randomly, without repetition, from the integers `0, ..., n - 1` (inclusive).
`long` `nextPoisson(double mean)` Generates a random value from the Poisson distribution with the given mean.
`Object[]` `nextSample(Collection<?> c, int k)` Returns an array of `k` objects selected randomly from the Collection `c`.
`String` `nextSecureHexString(int len)` Generates a random string of hex characters from a secure random sequence.
`int` `nextSecureInt(int lower, int upper)` Generates a uniformly distributed random integer between `lower` and `upper` (endpoints included) from a secure random sequence.
`long` `nextSecureLong(long lower, long upper)` Generates a uniformly distributed random long integer between `lower` and `upper` (endpoints included) from a secure random sequence.
`double` `nextT(double df)` Generates a random value from the `T Distribution`.
`double` `nextUniform(double lower, double upper)` Generates a uniformly distributed random value from the open interval `(lower, upper)` (i.e., endpoints excluded).
`double` `nextUniform(double lower, double upper, boolean lowerInclusive)` Generates a uniformly distributed random value from the interval `(lower, upper)` or the interval `[lower, upper)`.
`double` `nextWeibull(double shape, double scale)` Generates a random value from the `Weibull Distribution`.
`int` `nextZipf(int numberOfElements, double exponent)` Generates a random value from the `Zipf Distribution`.
`void` `reSeed()` Reseeds the random number generator with `System.currentTimeMillis() + System.identityHashCode(this))`.
`void` `reSeed(long seed)` Reseeds the random number generator with the supplied seed.
`void` `reSeedSecure()` Reseeds the secure random number generator with the current time in milliseconds.
`void` `reSeedSecure(long seed)` Reseeds the secure random number generator with the supplied seed.
`void` `setSecureAlgorithm(String algorithm, String provider)` Sets the PRNG algorithm for the underlying SecureRandom instance using the Security Provider API.
`private void` `shuffle(int[] list, int end)` Uses a 2-cycle permutation shuffle to randomly re-order the last elements of list.

### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

### Field Detail

#### serialVersionUID

    private static final long serialVersionUID

Serializable version identifier

See Also:
  Constant Field Values

#### rand

    private RandomGenerator rand

underlying random number generator

#### secRand

    private SecureRandom secRand

underlying secure random number generator

### Constructor Detail

#### RandomDataGenerator

    public RandomDataGenerator()

Construct a RandomDataGenerator, using a default random generator as the source of randomness.

The default generator is a `Well19937c` seeded with `System.currentTimeMillis() + System.identityHashCode(this))`. The generator is initialized and seeded on first use.

#### RandomDataGenerator

    public RandomDataGenerator(RandomGenerator rand)

Construct a RandomDataGenerator using the supplied `RandomGenerator` as the source of (non-secure) random data.

Parameters:
  `rand` - the source of (non-secure) random data (may be null, resulting in the default generator)

### Method Detail

#### nextHexString

    public String nextHexString(int len)
                         throws NotStrictlyPositiveException

Generates a random string of hex characters of length `len`.

The generated string will be random, but not cryptographically secure. To generate cryptographically secure strings, use `RandomData.nextSecureHexString(int)`.

Algorithm Description: hex strings are generated using a 2-step process.

1. `len / 2 + 1` binary bytes are generated using the underlying Random
2. Each binary byte is translated into 2 hex digits

Specified by:
  `nextHexString` in interface `RandomData`
Parameters:
  `len` - the desired string length.
Returns:
  the random string.
Throws:
  `NotStrictlyPositiveException` - if `len <= 0`.

#### nextInt

    public int nextInt(int lower,
              int upper)
                throws NumberIsTooLargeException

Generates a uniformly distributed random integer between `lower` and `upper` (endpoints included).

The generated integer will be random, but not cryptographically secure. To generate cryptographically secure integer sequences, use `RandomData.nextSecureInt(int, int)`.

Specified by:
  `nextInt` in interface `RandomData`
Parameters:
  `lower` - lower bound for generated integer
  `upper` - upper bound for generated integer
Returns:
  a random integer greater than or equal to `lower` and less than or equal to `upper`
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`

#### nextLong

    public long nextLong(long lower,
                long upper)
                  throws NumberIsTooLargeException

Generates a uniformly distributed random long integer between `lower` and `upper` (endpoints included).

The generated long integer values will be random, but not cryptographically secure. To generate cryptographically secure sequences of longs, use `RandomData.nextSecureLong(long, long)`.

Specified by:
  `nextLong` in interface `RandomData`
Parameters:
  `lower` - lower bound for generated long integer
  `upper` - upper bound for generated long integer
Returns:
  a random long integer greater than or equal to `lower` and less than or equal to `upper`
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`

#### nextSecureHexString

    public String nextSecureHexString(int len)
                               throws NotStrictlyPositiveException

Generates a random string of hex characters from a secure random sequence.

If cryptographic security is not required, use `RandomData.nextHexString(int)`.

Algorithm Description: hex strings are generated in 40-byte segments using a 3-step process.

1. 20 random bytes are generated using the underlying `SecureRandom`.
2. SHA-1 hash is applied to yield a 20-byte binary digest.
3. Each byte of the binary digest is converted to 2 hex digits.

Specified by:
  `nextSecureHexString` in interface `RandomData`
Parameters:
  `len` - the length of the string to be generated
Returns:
  a random string of hex characters of length `len`
Throws:
  `NotStrictlyPositiveException` - if `len <= 0`

#### nextSecureInt

    public int nextSecureInt(int lower,
                    int upper)
                      throws NumberIsTooLargeException

Generates a uniformly distributed random integer between `lower` and `upper` (endpoints included) from a secure random sequence.

Sequences of integers generated using this method will be cryptographically secure. If cryptographic security is not required, `RandomData.nextInt(int, int)` should be used instead of this method.

Definition: Secure Random Sequence

Specified by:
  `nextSecureInt` in interface `RandomData`
Parameters:
  `lower` - lower bound for generated integer
  `upper` - upper bound for generated integer
Returns:
  a random integer greater than or equal to `lower` and less than or equal to `upper`.
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`.

#### nextSecureLong

    public long nextSecureLong(long lower,
                      long upper)
                        throws NumberIsTooLargeException

Generates a uniformly distributed random long integer between `lower` and `upper` (endpoints included) from a secure random sequence.

Sequences of long values generated using this method will be cryptographically secure. If cryptographic security is not required, `RandomData.nextLong(long, long)` should be used instead of this method.

Definition: Secure Random Sequence

Specified by:
  `nextSecureLong` in interface `RandomData`
Parameters:
  `lower` - lower bound for generated integer
  `upper` - upper bound for generated integer
Returns:
  a random long integer greater than or equal to `lower` and less than or equal to `upper`.
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`.

#### nextPoisson

    public long nextPoisson(double mean)
                     throws NotStrictlyPositiveException

Generates a random value from the Poisson distribution with the given mean.

Definition: Poisson Distribution

Algorithm Description:

- For small means, uses simulation of a Poisson process using Uniform deviates, as described here. The Poisson process (and hence value returned) is bounded by 1000 * mean.
- For large means, uses the rejection algorithm described in
  Devroye, Luc. (1981).The Computer Generation of Poisson Random Variables Computing vol. 26 pp. 197-207.

Specified by:
  `nextPoisson` in interface `RandomData`
Parameters:
  `mean` - the mean of the Poisson distribution
Returns:
  a random value following the specified Poisson distribution
Throws:
  `NotStrictlyPositiveException` - if `len <= 0`

#### nextGaussian

    public double nextGaussian(double mu,
                      double sigma)
                        throws NotStrictlyPositiveException

Generates a random value from the Normal (or Gaussian) distribution with specified mean and standard deviation.

Definition: Normal Distribution

Specified by:
  `nextGaussian` in interface `RandomData`
Parameters:
  `mu` - the mean of the distribution
  `sigma` - the standard deviation of the distribution
Returns:
  a random value following the specified Gaussian distribution
Throws:
  `NotStrictlyPositiveException` - if `sigma <= 0`.

#### nextExponential

    public double nextExponential(double mean)
                           throws NotStrictlyPositiveException

Generates a random value from the exponential distribution with specified mean.

Definition: Exponential Distribution

Algorithm Description: Uses the Algorithm SA (Ahrens) from p. 876 in: [1]: Ahrens, J. H. and Dieter, U. (1972). Computer methods for sampling from the exponential and normal distributions. Communications of the ACM, 15, 873-882.

Specified by:
  `nextExponential` in interface `RandomData`
Parameters:
  `mean` - the mean of the distribution
Returns:
  a random value following the specified exponential distribution
Throws:
  `NotStrictlyPositiveException` - if `mean <= 0`.

#### nextGamma

    public double nextGamma(double shape,
                   double scale)
                     throws NotStrictlyPositiveException

Generates a random value from the `Gamma Distribution`.

This implementation uses the following algorithms:

For 0 < shape < 1:
Ahrens, J. H. and Dieter, U., Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing, 12, 223-246, 1974.

For shape >= 1:
Marsaglia and Tsang, A Simple Method for Generating Gamma Variables. ACM Transactions on Mathematical Software, Volume 26 Issue 3, September, 2000.

Parameters:
  `shape` - the median of the Gamma distribution
  `scale` - the scale parameter of the Gamma distribution
Returns:
  random value sampled from the Gamma(shape, scale) distribution
Throws:
  `NotStrictlyPositiveException` - if `shape <= 0` or `scale <= 0`.

#### nextHypergeometric

    public int nextHypergeometric(int populationSize,
                         int numberOfSuccesses,
                         int sampleSize)
                           throws NotPositiveException,
                                  NotStrictlyPositiveException,
                                  NumberIsTooLargeException

Generates a random value from the `Hypergeometric Distribution`.

Parameters:
  `populationSize` - the population size of the Hypergeometric distribution
  `numberOfSuccesses` - number of successes in the population of the Hypergeometric distribution
  `sampleSize` - the sample size of the Hypergeometric distribution
Returns:
  random value sampled from the Hypergeometric(numberOfSuccesses, sampleSize) distribution
Throws:
  `NumberIsTooLargeException` - if `numberOfSuccesses > populationSize`, or `sampleSize > populationSize`.
  `NotStrictlyPositiveException` - if `populationSize <= 0`.
  `NotPositiveException` - if `numberOfSuccesses < 0`.

#### nextPascal

    public int nextPascal(int r,
                 double p)
                   throws NotStrictlyPositiveException,
                          OutOfRangeException

Generates a random value from the `Pascal Distribution`.

Parameters:
  `r` - the number of successes of the Pascal distribution
  `p` - the probability of success of the Pascal distribution
Returns:
  random value sampled from the Pascal(r, p) distribution
Throws:
  `NotStrictlyPositiveException` - if the number of successes is not positive
  `OutOfRangeException` - if the probability of success is not in the range `[0, 1]`.

#### nextT

    public double nextT(double df)
                 throws NotStrictlyPositiveException

Generates a random value from the `T Distribution`.

Parameters:
  `df` - the degrees of freedom of the T distribution
Returns:
  random value from the T(df) distribution
Throws:
  `NotStrictlyPositiveException` - if `df <= 0`

#### nextWeibull

    public double nextWeibull(double shape,
                     double scale)
                       throws NotStrictlyPositiveException

Generates a random value from the `Weibull Distribution`.

Parameters:
  `shape` - the shape parameter of the Weibull distribution
  `scale` - the scale parameter of the Weibull distribution
Returns:
  random value sampled from the Weibull(shape, size) distribution
Throws:
  `NotStrictlyPositiveException` - if `shape <= 0` or `scale <= 0`.

#### nextZipf

    public int nextZipf(int numberOfElements,
               double exponent)
                 throws NotStrictlyPositiveException

Generates a random value from the `Zipf Distribution`.

Parameters:
  `numberOfElements` - the number of elements of the ZipfDistribution
  `exponent` - the exponent of the ZipfDistribution
Returns:
  random value sampled from the Zipf(numberOfElements, exponent) distribution
Throws:
  `NotStrictlyPositiveException` - if `numberOfElements <= 0` or `exponent <= 0`.

#### nextBeta

    public double nextBeta(double alpha,
                  double beta)

Generates a random value from the `Beta Distribution`.

Parameters:
  `alpha` - first distribution shape parameter
  `beta` - second distribution shape parameter
Returns:
  random value sampled from the beta(alpha, beta) distribution

#### nextBinomial

    public int nextBinomial(int numberOfTrials,
                   double probabilityOfSuccess)

Generates a random value from the `Binomial Distribution`.

Parameters:
  `numberOfTrials` - number of trials of the Binomial distribution
  `probabilityOfSuccess` - probability of success of the Binomial distribution
Returns:
  random value sampled from the Binomial(numberOfTrials, probabilityOfSuccess) distribution

#### nextCauchy

    public double nextCauchy(double median,
                    double scale)

Generates a random value from the `Cauchy Distribution`.

Parameters:
  `median` - the median of the Cauchy distribution
  `scale` - the scale parameter of the Cauchy distribution
Returns:
  random value sampled from the Cauchy(median, scale) distribution

#### nextChiSquare

    public double nextChiSquare(double df)

Generates a random value from the `ChiSquare Distribution`.

Parameters:
  `df` - the degrees of freedom of the ChiSquare distribution
Returns:
  random value sampled from the ChiSquare(df) distribution

#### nextF

    public double nextF(double numeratorDf,
               double denominatorDf)
                 throws NotStrictlyPositiveException

Generates a random value from the `F Distribution`.

Parameters:
  `numeratorDf` - the numerator degrees of freedom of the F distribution
  `denominatorDf` - the denominator degrees of freedom of the F distribution
Returns:
  random value sampled from the F(numeratorDf, denominatorDf) distribution
Throws:
  `NotStrictlyPositiveException` - if `numeratorDf <= 0` or `denominatorDf <= 0`.

#### nextUniform

    public double nextUniform(double lower,
                     double upper)
                       throws NumberIsTooLargeException,
                              NotFiniteNumberException,
                              NotANumberException

Generates a uniformly distributed random value from the open interval `(lower, upper)` (i.e., endpoints excluded).

Definition: Uniform Distribution `lower` and `upper - lower` are the location and scale parameters, respectively.

Algorithm Description: scales the output of Random.nextDouble(), but rejects 0 values (i.e., will generate another random double if Random.nextDouble() returns 0). This is necessary to provide a symmetric output interval (both endpoints excluded).

Specified by:
  `nextUniform` in interface `RandomData`
Parameters:
  `lower` - the exclusive lower bound of the support
  `upper` - the exclusive upper bound of the support
Returns:
  a uniformly distributed random value between lower and upper (exclusive)
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`
  `NotFiniteNumberException` - if one of the bounds is infinite
  `NotANumberException` - if one of the bounds is NaN

#### nextUniform

    public double nextUniform(double lower,
                     double upper,
                     boolean lowerInclusive)
                       throws NumberIsTooLargeException,
                              NotFiniteNumberException,
                              NotANumberException

Generates a uniformly distributed random value from the interval `(lower, upper)` or the interval `[lower, upper)`. The lower bound is thus optionally included, while the upper bound is always excluded.

Definition: Uniform Distribution `lower` and `upper - lower` are the location and scale parameters, respectively.

Algorithm Description: if the lower bound is excluded, scales the output of Random.nextDouble(), but rejects 0 values (i.e., will generate another random double if Random.nextDouble() returns 0). This is necessary to provide a symmetric output interval (both endpoints excluded).

Specified by:
  `nextUniform` in interface `RandomData`
Parameters:
  `lower` - the lower bound of the support
  `upper` - the exclusive upper bound of the support
  `lowerInclusive` - `true` if the lower bound is inclusive
Returns:
  uniformly distributed random value in the `(lower, upper)` interval, if `lowerInclusive` is `false`, or in the `[lower, upper)` interval, if `lowerInclusive` is `true`
Throws:
  `NumberIsTooLargeException` - if `lower >= upper`
  `NotFiniteNumberException` - if one of the bounds is infinite
  `NotANumberException` - if one of the bounds is NaN

#### nextPermutation

    public int[] nextPermutation(int n,
                        int k)
                          throws NumberIsTooLargeException,
                                 NotStrictlyPositiveException

Generates an integer array of length `k` whose entries are selected randomly, without repetition, from the integers `0, ..., n - 1` (inclusive).

Generated arrays represent permutations of `n` taken `k` at a time.

Uses a 2-cycle permutation shuffle. The shuffling process is described here.

Specified by:
  `nextPermutation` in interface `RandomData`
Parameters:
  `n` - the domain of the permutation
  `k` - the size of the permutation
Returns:
  a random `k`-permutation of `n`, as an array of integers
Throws:
  `NumberIsTooLargeException` - if `k > n`.
  `NotStrictlyPositiveException` - if `k <= 0`.

#### nextSample

    public Object[] nextSample(Collection<?> c,
                      int k)
                        throws NumberIsTooLargeException,
                               NotStrictlyPositiveException

Returns an array of `k` objects selected randomly from the Collection `c`.

Sampling from `c` is without replacement; but if `c` contains identical objects, the sample may include repeats. If all elements of `c` are distinct, the resulting object array represents a Simple Random Sample of size `k` from the elements of `c`.

Algorithm Description: Uses a 2-cycle permutation shuffle to generate a random permutation of `c.size()` and then returns the elements whose indexes correspond to the elements of the generated permutation. This technique is described, and proven to generate random samples here

Specified by:
  `nextSample` in interface `RandomData`
Parameters:
  `c` - the collection to be sampled
  `k` - the size of the sample
Returns:
  a random sample of `k` elements from `c`
Throws:
  `NumberIsTooLargeException` - if `k > c.size()`.
  `NotStrictlyPositiveException` - if `k <= 0`.

#### reSeed

    public void reSeed(long seed)

Reseeds the random number generator with the supplied seed.

Will create and initialize if null.

Parameters:
  `seed` - the seed value to use

#### reSeedSecure

    public void reSeedSecure()

Reseeds the secure random number generator with the current time in milliseconds.

Will create and initialize if null.

#### reSeedSecure

    public void reSeedSecure(long seed)

Reseeds the secure random number generator with the supplied seed.

Will create and initialize if null.

Parameters:
  `seed` - the seed value to use

#### reSeed

    public void reSeed()

Reseeds the random number generator with `System.currentTimeMillis() + System.identityHashCode(this))`.

#### setSecureAlgorithm

    public void setSecureAlgorithm(String algorithm,
                          String provider)
                            throws NoSuchAlgorithmException,
                                   NoSuchProviderException

Sets the PRNG algorithm for the underlying SecureRandom instance using the Security Provider API. The Security Provider API is defined in Java Cryptography Architecture API Specification & Reference.

USAGE NOTE: This method carries significant overhead and may take several seconds to execute.

Parameters:
  `algorithm` - the name of the PRNG algorithm
  `provider` - the name of the provider
Throws:
  `NoSuchAlgorithmException` - if the specified algorithm is not available
  `NoSuchProviderException` - if the specified provider is not installed

#### getRan

    private RandomGenerator getRan()

Returns the RandomGenerator used to generate non-secure random data.

Creates and initializes a default generator if null. Uses a `Well19937c` generator with `System.currentTimeMillis() + System.identityHashCode(this))` as the default seed.

Returns:
  the Random used to generate random data

#### initRan

    private void initRan()

Sets the default generator to a `Well19937c` generator seeded with `System.currentTimeMillis() + System.identityHashCode(this))`.

#### getSecRan

    private SecureRandom getSecRan()

Returns the SecureRandom used to generate secure random data.

Creates and initializes if null. Uses `System.currentTimeMillis() + System.identityHashCode(this)` as the default seed.

Returns:
  the SecureRandom used to generate secure random data

#### shuffle

    private void shuffle(int[] list,
               int end)

Uses a 2-cycle permutation shuffle to randomly re-order the last elements of list.

Parameters:
  `list` - list to be shuffled
  `end` - element past which shuffling begins

#### getNatural

    private int[] getNatural(int n)

Returns an array representing n.

Parameters:
  `n` - the natural number to represent
Returns:
  array with entries = elements of n