Abstract not found

Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudo-random number generation, i.e. for 3-term linear recurrences dened by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15. 1.

Abstract. The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm. If 2 r − 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1.

Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there is no irreducible trinomial of degree r.

. We improve on a lattice algorithm of Tezuka for the computation of the k-distribution of a class of random number generators based on finite fields. We show how this is applied to the problem of constructing, for such generators, an output mapping yielding optimal k-distribution. 1. Introduction Extensive classes of random number generators have the following structure. The state space is a finite field F of characteristic 2. We denote by d its degree over F 2 , and sometimes refer to it as the order of the generator. Any state y 2 F evolves into a state xy, where the distinguished element, x 2 F , completely determines the evolution of the generator. Finally, the generator in state y outputs a w-bit vector \Phi(y) = (OE(y 1 y); : : : ; OE(y w y)) 2 F 2 w , where OE : F ! F 2 is any non-zero linear form over F 2 , and where y 1 ; : : : ; y l are suitably chosen non-zero elements of F . The study of the k-distribution of the output sequence involves the computation, for all l w an...

Pseudo-random numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime. These generators are based on "almost primitive" trinomials, that is trinomials having a large primitive factor. They enable very fast vector/parallel implementations with excellent statistical properties.

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r 2) bit-operations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bit-operations than the standard algorithm. Using the algorithm, we have found 18 new irreducible trinomials of degree r in the range 100151 ≤ r ≤ 700057. If r is a Mersenne exponent (i.e. 2 r −1 is a Mersenne prime), then an irreducible trinomial is primitive. Primitive trinomials are of interest because they can be used to give pseudorandom number generators with period at least 2 r − 1. We give examples of primitive trinomials for r = 756839, 859433, and 3021377. The three results for r = 756839 are new. The results for r = 859433 extend and correct some computations of Kumada et al. [Math. Comp. 69 (2000), 811–814]. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1 Copyright c○2000, the authors. rpb199tr typeset using L ATEX 1 1

Abstract. Fast and reliable pseudo-random number generators are required for simulation and other applications in Scientific Computing. We outline the requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations with excellent statistical properties. We also discuss the problem of initialising random number generators, and consider how to combine two or more generators to give a better (though usually slower) generator. 1

We give a new algorithm for performing the distinct-degree factorization of a polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our approach is that a finer level of blocking replaces multiplications by squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval polynomials when P(x) is sparse. As an application we give a fast algorithm to search for all irreducible trinomials x r + x s + 1 of degree r over GF(2), while producing a certificate that can be checked in less time than the full search. Naive algorithms cost O(r 2) per trinomial, thus O(r 3) to search over all trinomials of given degree r. Under a plausible assumption about the distribution of factors of trinomials, the new algorithm has complexity O(r 2 (log r) 3/2 (log log r) 1/2) for the search over all trinomials of degree r. Our implementation achieves a speedup of greater than a factor of 560 over the naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our program, we have found two new primitive trinomials of degree 24036583 over GF(2) (the previous record degree was 6972593).