PollardRhoBrentMontgomery64 (org.matheclipse:matheclipse-gpl 3.0.0 API)

java.lang.Object
- de.tilman_neumann.jml.factor.FactorAlgorithm
- - de.tilman_neumann.jml.factor.pollardRho.PollardRhoBrentMontgomery64

```
public class PollardRhoBrentMontgomery64
extends FactorAlgorithm
```
Brents's improvement of Pollard's Rho algorithm using Montgomery multiplication. The main reason why Montgomery multiplication is helpful for Pollard-Rho is that no conversions to/from Montgomery form are required. In this implementation I managed to use the Montgomery reducer R=2^64, which simplifies the Montgomery multiplication a good deal. Another small performance improvement stems from using the polynomial x*(x+1) instead of x^2+c, which saves us the addition modulo N after each Montgomery multiplication.

Author:

Tilman Neumann

Field Summary
- Fields inherited from class de.tilman_neumann.jml.factor.FactorAlgorithm
  DEFAULT, NUM_PRIMES_FOR_31_BIT_TDIV, tdivLimit

Constructor Summary

Constructors
Constructor Description

PollardRhoBrentMontgomery64()

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method	Description
`long`	`findSingleFactor(long N)`
`BigInteger`	`findSingleFactor(BigInteger N)`	Find a single factor of the given N, which is composite and odd.
`String`	`getName()`
`static void`	`main(String[] args)`	Test.
`static long`	`montMul64(long a, long b, long N, long Nhat)`	Montgomery multiplication of a*b mod n.

Methods inherited from class de.tilman_neumann.jml.factor.FactorAlgorithm
factor, factor, searchFactors

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - PollardRhoBrentMontgomery64
```
public PollardRhoBrentMontgomery64()
```
- Method Detail
  - getName
```
public String getName()
```
    Specified by:
    
    getName in class FactorAlgorithm
    
    Returns:
    
    The name of the algorithm, possibly including important parameters.
  - findSingleFactor
```
public BigInteger findSingleFactor(BigInteger N)
```
    Description copied from class: FactorAlgorithm
    
    Find a single factor of the given N, which is composite and odd.
    
    Specified by:
    
    findSingleFactor in class FactorAlgorithm
    
    Returns:
    
    factor
  - findSingleFactor
```
public long findSingleFactor(long N)
```
  - montMul64
```
public static long montMul64(long a,
                             long b,
                             long N,
                             long Nhat)
```
    Montgomery multiplication of a*b mod n. ("mulredcx" in Yafu)
    
    Parameters:
    
    a -
    
    b -
    
    N -
    
    Nhat - complement of N mod 2^64
    
    Returns:
    
    Montgomery multiplication of a*b mod n
  - main
```
public static void main(String[] args)
```
    Test. Test numbers: 3225275494496681 (52 bits) = 56791489 * 56791529 322527333642009919 (59 bits) = 567914891 * 567914909 3225273260887418687 (62 bits) = 567914891 * 5679148957
    
    Parameters:
    
    args - ignored