2⁶⁴ -2¹⁰ + 1 Modulus Linear Congruential Generator

Due to the discovery of a pattern for random number generators based on Mersenne primes, a family of prime modulus generators has been explored that removes this pattern while still maintaining good efficiency for modular arithmetic. In addition to removing the offending pattern, these generators also offer a slightly longer period because they use more of the available 64-bits in the random number state. As discussed in [5], the key to removing the pattern resulting from the Mersenne prime modular arithmetic is using special primes with a ``+1'' instead of ``-1'' as the last term in the expression composed of powers of two. The first of these is a prime modulus linear congruential generator provided by the recurrence relation:

x[n] = a x[n - 1]   (modp) (7)

Where the multiplier, a = 3355703948966806693, is hardwired into the algorithm and p is the prime, 2⁶⁴ -2¹⁰ + 1. The multiplier was again chosen to obtain maximal period of the generator, 2⁶⁴ -2¹⁰ $\approx$ 1.8 x 10¹⁹.

The interface routines for this generator are declared as: Rng_Type RngP64_10Seed( Rng_UInt32 i, Rng_UInt32 j );

Rng_Type RngP64_10Spawn( Rng_Type *x );

int iRngP64_10( Rng_Type *x );

double dRngP64_10( Rng_Type *x );

float fRngP64_10( Rng_Type *x );

This random number generator suffers from no patterns that we are aware of. We are currently evaluating its quality using statistical tests. The fact that this generator has a prime modulus and its period is close to 2⁶⁴ makes it very attractive. This generator also traps a zeroed state, making it equally useful for debugging.