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On the period of the linear congruential and power generators
 Acta Arith
"... 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 ..."
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Cited by 8 (3 self)
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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
On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators
 MATH. COMP
, 2010
"... 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 ..."
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Cited by 2 (2 self)
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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.
On some dynamical systems in finite fields and residue rings
 Discr. and Cont.Dynam.Syst.,Ser.A
"... 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 t ..."
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Cited by 2 (2 self)
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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
PRIME CHAINS AND PRATT TREES
, 2009
"... 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 ..."
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Cited by 1 (0 self)
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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 ElliottHalberstam 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.
Compositions with the Euler and Carmichael Functions
"... 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 ..."
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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
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A25 IRREDUCIBLE RADICAL EXTENSIONS AND EULERFUNCTION CHAINS
"... We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is ϕ(m), where m is the smallest multiple of n divisible by each prim ..."
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We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is ϕ(m), where m is the smallest multiple of n divisible by each prime factor of ϕ(m). The prime factors of m/n are precisely the primes not dividing n but which do divide some number in the “Euler chain ” ϕ(n), ϕ(ϕ(n)),.... For each fixed k, we show that D(n)> n k on a set of asymptotic density 1. 1.
LOWER BOUNDS ON THE PERIOD OF SOME PSEUDORANDOM NUMBER GENERATORS
"... We are interested in obtaining lower bounds on the periods of two standard pseudorandom number generators from number theorythe linear congruential generator, rst introduced by D. H. Lehmer, and the so called power generator. For the former, given integers e; b; n (with e; n> 1) and a ..."
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We are interested in obtaining lower bounds on the periods of two standard pseudorandom number generators from number theorythe linear congruential generator, rst introduced by D. H. Lehmer, and the so called power generator. For the former, given integers e; b; n (with e; n> 1) and a