Results 1  10
of
15
Random Generators and Normal Numbers
 EXPERIMENTAL MATHEMATICS
, 2000
"... Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental ..."
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Cited by 25 (11 self)
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Pursuant to the authors' previous chaoticdynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2normal (i.e. normal to base 2), we are able to establish bnormality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googolth (i.e. 10 100  th) binary bit of 2;3 is 0. We also present a collection of other results  such as density results and irrationality proofs based on PRNG ideas  for various special numbers.
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 7 (2 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
The iterated Carmichael λ function and the number of cycles of the power generator
, 2005
"... A common pseudorandom number generator is the power generator: x ↦ → x ℓ (mod n). Here, ℓ, n are fixed integers at least 2, and one constructs a pseudorandom sequence by starting at some residue mod n and iterating this ℓth power map. (Because it is the easiest to compute, one often takes ℓ = 2; thi ..."
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Cited by 6 (2 self)
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A common pseudorandom number generator is the power generator: x ↦ → x ℓ (mod n). Here, ℓ, n are fixed integers at least 2, and one constructs a pseudorandom sequence by starting at some residue mod n and iterating this ℓth power map. (Because it is the easiest to compute, one often takes ℓ = 2; this case is known as the BBS generator, for Blum,
Squarefree Values of the Carmichael Function
 J. NUM. THEORY
, 2003
"... We obtain an asymptotic formula for the number of squarefree values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of squarefree values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ðk þ oð1ÞÞ x ln 1 a x; where a 0:37395y is the Artin ..."
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Cited by 5 (3 self)
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We obtain an asymptotic formula for the number of squarefree values among p 1; for primes ppx; and we apply it to derive the following asymptotic formula for LðxÞ; the number of squarefree values of the Carmichael function lðnÞ for 1pnpx; LðxÞ ðk þ oð1ÞÞ x ln 1 a x; where a 0:37395y is the Artin constant, and k 0:80328y is another absolute constant.
Average Multiplicative Orders of Elements Modulo n
 Acta Arith
"... We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary ..."
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Cited by 4 (1 self)
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We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary 20K01 1
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
On the uniformity of distribution of the RSA pairs
 Math. Comp
"... Abstract. Let m = pl be a product of two distinct primes p and l. Weshow that for almost all exponents e with gcd(e, ϕ(m)) = 1 the RSA pairs (x, xe) are uniformly distributed modulo m when x runs through • the group of units Z ∗ m modulo m (that is, as in the classical RSA scheme); • the set of kp ..."
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Cited by 2 (1 self)
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Abstract. Let m = pl be a product of two distinct primes p and l. Weshow that for almost all exponents e with gcd(e, ϕ(m)) = 1 the RSA pairs (x, xe) are uniformly distributed modulo m when x runs through • the group of units Z ∗ m modulo m (that is, as in the classical RSA scheme); • the set of kproducts x = ai1 ···ai, 1 ≤ i1 < ·· · < ik ≤ n, where k a1, ·· ·,an ∈ Z ∗ m are selected at random (that is, as in the recently introduced RSA scheme with precomputation). These results are based on some new bounds of exponential sums. 1.
CORRIGENDUM TO “PERIOD OF THE POWER GENERATOR AND SMALL VALUES OF CARMICHAEL’S FUNCTION”
"... We are indebted to Kelly Postelmans whose question drew our attention to a slip in the proof of Theorem 8 of [1]. In particular, we asserted that for a fixed number n, the number of pairs of primes p, l with gcd(p − 1,l − 1)
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Cited by 2 (0 self)
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We are indebted to Kelly Postelmans whose question drew our attention to a slip in the proof of Theorem 8 of [1]. In particular, we asserted that for a fixed number n, the number of pairs of primes p, l with gcd(p − 1,l − 1) <Dand λ(λ(pl)) = n is at most Dτ(n), an assertion which now seems unjustified. (The notation is defined below.) In this note we give a corrected proof of Theorem 8. As in [1] we consider the power generator (1) un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., with the initial value u0 = ϑ (an integer coprime to m) andexponent e (an integer at least 2). We recall that for an integer n ≥ 1theCarmichael function λ(n) is the largest order occurring amongst elements of the unit group in the residue ring modulo n. As usual, ϕ denotes Euler’s function. We let τ(n) denote the number of natural divisors of n, weletω(n) denote the number of divisors of n that are prime, and we let Ω(n) denote the number of divisors of n that are (either a prime or) a prime power. An integer n is said to be squarefull if for each prime pn we
On the linear complexity profile of the power generator
 IEEE Trans. Inf. Theory
, 1998
"... Abstract We obtain a lower bound on the linear complexity profile of the power generator of pseudorandom numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the BlumBlumShub generator. In particular, these results imply that lattice redu ..."
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Cited by 1 (0 self)
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Abstract We obtain a lower bound on the linear complexity profile of the power generator of pseudorandom numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the BlumBlumShub generator. In particular, these results imply that lattice reduction attacks on such generators are not feasible.