We consider two standard pseudorandom number generators from number theory: the linear congruential generator and the power generator. For the former, we are given integers e, b, n (with e, n> 1) and a seed u0, and we compute the sequence

We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning. 1

In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degree growth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.

ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the Elliott-Halberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x. 1.

Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1