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

Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presented, including a synthesis algorithm (analogous to the Berlekamp-Massey algorithm for LFSR’s) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the Marsaglia-Zaman random number generator. Possible variations on the FCSR architecture are indicated at the end. Index Terms – Binary sequence, shift register, stream cipher, combiner with memory, cryptanalysis, 2-adic numbers, arithmetic code, 1/q sequence, linear span. 1

Pseudorandom sequences, with a variety of statistical properties (such as high linear span, low autocorrelation and pairwise cross-correlation values, and high pairwise hamming distance) are important in many areas of communications and computing (such as cryptography, spread spectrum communications, error correcting codes, and Monte Carlo integration). Binary sequences~ such as m-sequences, more general nonlinear feedback shift register sequences, and summation combiner sequences, have been widely studied by many researchers. Linear feedback shift register hardware can be used to relate certain of these sequences (such as m-sequences) to error correcting codes (such as first order Reed-Muller codes). In this paper a new type of feedback register, feedback with carry shift registers (or FCSRs), will be presented. These relatively simple devices can be used to relate summation combiner sequences, arithmetic codes, and 1/q sequences. We describe an algebraic framework, based on algebra over the 2-adic numbers, in which the sequences generated by FCSRs can be analyzed, in much the same way that algebra over finite fields can be used to analyze LFSR sequences. As a consequence of this analysis, we present a method for cracking the summation combiner [9] which has been suggested for generating cryptographicaily secure binary sequences. In general,

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.

Introduction 1.1. A pseudorox"q number gener ator (RNG) for high speed simulation and Monte CarS integrSqKx should have sever" pr" er"US : (1) it should haveenor""x perz d, (2) it should e hibitunifor distrqS""xI of d-tuples(for all d), (3) it should exhibit a good lattice str""Ezx in high dimensions, and (4) it should be e#ciently computable(prablexzF with a base b which is a power of 2). Typically the RNG is a member of a family ofsimilar generrxI withdi#erq tparU"xIEU and one hopes that parKq"qxI and seeds may be easily chosen so as toguarF tee pr" er"E" (1), (2), (3) and (4). Ther is no known family of RNG with all four pr" er"KS (see,for example, [M1]). 1.2. In [MZ], Mar aglia and Zaman showed that their add-with-carc (AWC) gener ator satisfy condition (1). By giving up on (4) and using an appr"FxIE" base b, they achieve good distrxSKEKx pr" er"Kq of d-tuplesfor values d wh

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

PRNGlib provides several pseudo-random number generators through a common interface on any Shared or Distributed Memory Parallel architecture. Common routines are specified to initialize the generators with appropriate seeds on each processor and to generate uniform or (normal, Poisson, exponential) distributed random vectors. We concentrate on those generators which assure high quality (i.e., passing most of the empirical and theoretical tests), have a long period, and can be calculated quickly, also in parallel, i.e., it must be possible to generate the same random sequence independent of the number of processors. This splitting facility implies a method to skip over n pseudo-random numbers without calculating all intermediate values, i.e., an O(log n) algorithm is required. Taking into account these criteria Lagged Fibonacci, Generalized Shift Register, and Multiplicative Linear Congruential generators are implemented with (almost) arbitrary specifications for lags, multipliers, m...

Using an extensive data set of 15,600 CDS and CDO tranche spreads on the North American Investment Grade CDX index I conduct an empirical analysis of a Duffie and Gârleanu (2001) intensitybased model for correlated defaults. I examine the model with respect to model assumptions, pricing in both the cross section and time series dimension, and hedging ability. The results show that the model assumptions are reasonable and that average prices are matched well. In addition, the model accurately tracks the prices over time of the more risky tranches. Finally, the model sensitivity of the most risky tranches to underlying CDS spreads match actual sensitivities better than those implied by the commonly used Gaussian copula. The last result suggests that the model is well-suited for hedging the equity tranche.

A d-feedback-with-carry shift register (d-FCSR) is a finite state machine, similar to a linear feedback shift register, in which a small amount of memory and a delay (by d-clock cycles) is used in the feedback algorithm (see [4, 5]). The output sequences of these simple devices may be described using arithmetic in a ramified extension field of the rational numbers. In this paper we show how many of these sequences may also be described using simple integer arithmetic, and consequently how to find such sequences with large periods. We also analyze the "arithmetic cross-correlation" between pairs of these sequences and show that it often vanishes identically. Finally we study the distribution properties of short sub-sequences of a d-FCSR sequence.

Abstract—We show that the random number generator of Marsaglia and Zaman produces the successive digits of a rational-adic number. (The-adic number system generalizes-adic numbers to an arbitrary integer base.) Using continued fractions, we derive an efficient prediction algorithm for this generator. Index Terms — Continued fractions, inductive inference,-adic numbers, pseudorandom sequences.