On the lattice structure of the add-with-carry and subtract-with-borrow random number generators

  title={On the lattice structure of the add-with-carry and subtract-with-borrow random number generators},
  author={Shu Tezuka and Pierre L'Ecuyer and Raymond Couture},
  journal={ACM Trans. Model. Comput. Simul.},
Marsaglia and Zaman recently proposed new classes of random number generators, called add-with-carry(AWC) and subtract-with-borrow(SWB), which are capable of quickly generating very long-period (pseudo)-random number sequences using very little memory. We show that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli. So, the AWC/SWB generators can be viewed as efficient ways of implementing such large linear congruential generators. As a… Expand
Linear recurrences with carry as uniform random number generators
In order to improve the uniformity of the d-dimensional distribution of the output of these subgenerators over their full period, a method for finding good parameters in terms of the spectral test is proposed. Expand
A revision of the subtract-with-borrow random number generators
The most popular and widely used subtract-with-borrow generator, also known as RANLUX, is reimplemented as a linear congruential generator using large integer arithmetic with the modulus size of 576 bits to show a significant gain in generation speed. Expand
On the lattice structure of certain linear congruential sequences related to AWC/SWB generators
We analyze the lattice structure of certain types of linear congru- ential generators (LCGs), which include close approximations to the add-with- carry and subtract-with-borrow (AWC/SWB) randomExpand
Spectral Analysis of the MIXMAX Random Number Generators
It is shown that for coordinates at specific lags not too far apart, in three dimensions, all the nonzero points lie in only two hyperplanes, reminiscent of the behavior of lagged-Fibonacci and AWC/SWB generators. Expand
There is no known family of RNG with all four properties (see, for example, [M1], but one hopes that parameters and seeds may be easily chosen so as to guarantee properties (1, (2), (3) and (4). Expand
Efficient multiply-with-carry random number generators with maximal period
A simple modification of the multiply-with-carry random number generators of Marsaglia and Couture and L'Écuyer is proposed, which are both efficient and exhibit maximal period. Expand
Sum-discrepancy test on pseudorandom number generators
We introduce a non-empirical test on pseudorandom number generators (prng), named sum-discrepancy test. We compute the distribution of the sum of consecutive m outputs of a prng to be tested, underExpand
Uniform random number generation
  • P. L'Ecuyer
  • Computer Science, Mathematics
  • Ann. Oper. Res.
  • 1994
Practical ways of generating uniform variates for several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear Congruential over fields of formal series, and combined generators are examined. Expand
By Zeckendorf’s theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can beExpand
Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences
  • P. L'Ecuyer
  • Computer Science, Mathematics
  • INFORMS J. Comput.
  • 1997
This article considers the case where the t values taken are not successive, but separated by lags that are chosen a priori, and gives lower bounds on the distance between hyperplanes. Expand


Analysis of add-with-carry and subtract-with-borrow generators
It is shown that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli, as well as the theoretical properties of such generators, namely via the spectral and lattice tests. Expand
Lattice structure of pseudorandom sequences from shift-register generators
  • S. Tezuka
  • Computer Science, Mathematics
  • 1990 Winter Simulation Conference Proceedings
  • 1990
The author develops a theory of the lattice structure of pseudorandom sequences from shift register generators, i.e. Tausworthe sequences and GFSR (generalized feedback shift register) sequences, and derives a theorem that links the k-distribution of such sequences and the successive minima of thek-dimensional lattice over GF(2,x) associated with the sequences, thereby leading to the geometric interpretation of the crust structure of these sequences. Expand
A random number generator for PC's
Abstract It is now possible to do serious scientific work on personal computers (PC's). Many simulation studies, whether exploratory or for production runs, call for random numbers. We offer here aExpand
Random numbers for simulation
This paper focuses mainly on efficient and recently proposed techniques for generating uniform pseudorandom numbers, and aims to design more robust generators without having to pay too much in terms of portability, flexibility, and efficiency. Expand
Matrices and the structure of random number sequences
Abstract This note provides a short, self-contained treatment, using linear algebra and matrix theory, for establishing maximal periods, underlying structure, and choice of starting values forExpand
A Review of Pseudorandom Number Generators
This is a brief review of the current situation concerning practical pseudorandom number generation for Monte Carlo calculations. The conclusion is that pseudorandom number generators with theExpand
The Art of Computer Programming
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. Expand
A Current View of Random Number Generation. Computer Science and Statistics, Proceedings of the Sixteenth Symposium on the Interface, Atlanta, march
  • Elsevier Science Publ. (North-Holland),
  • 1984
The Art of Computer Programming: Seminumerical Algorithms, vol. 2, second edition
  • 1981
A New Class of Random Number Generators