Results 1  10
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53
The Carmichael Numbers up to 10^15
, 1992
"... There are 105212 Carmichael numbers up to 10 : we describe the calculations. ..."
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There are 105212 Carmichael numbers up to 10 : we describe the calculations.
Period of the power generator and small values of Carmichael’s function
 Math.Comp.,70
"... Abstract. Consider the pseudorandom number generator un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., where we are given the modulus m, the initial value u0 = ϑ and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, l are different primes of the same ..."
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Cited by 18 (11 self)
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Abstract. Consider the pseudorandom number generator un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., where we are given the modulus m, the initial value u0 = ϑ and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p, l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = pl, if the period of the sequence (un) exceeds m3/4+ε, then the sequence is uniformly distributed. We show rigorously that for almost all choices of p, l it is the case that for almost all choices of ϑ, e, the period of the power generator exceeds (pl) 1−ε. And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to rulingout the cycling attack on the RSA cryptosystem and to socalled timerelease crypto. The principal tool is an estimate related to the Carmichael function λ(m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m. In particular, we show that for any ∆ ≥ (log log N) 3,wehave λ(m) ≥ N exp(−∆) for all integers m with 1 ≤ m ≤ N, apartfromatmost N exp −0.69 ( ∆ log ∆) 1/3) exceptions. 1.
Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 12 (1 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
Nagaraj, Density of Carmichael numbers with three prime factors
 Math.Comp.66 (1997), 1705–1708. MR 98d:11110
"... Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1. ..."
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Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1.
An asymptotic formula for the number of smooth values of a polynomial
 J. Number Theory
, 1999
"... Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from mult ..."
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Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity
Counting Curves and Their Projections
 Computational Complexity
, 1996
"... . Some deterministic and probabilistic methods are presented for counting and estimating the number of points on curves over finite fields, and on their projections. The classical question of estimating the size of the image of a univariate polynomial is a special case. For curves given by spars ..."
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. Some deterministic and probabilistic methods are presented for counting and estimating the number of points on curves over finite fields, and on their projections. The classical question of estimating the size of the image of a univariate polynomial is a special case. For curves given by sparse polynomials, the counting problem is #Pcomplete via probabilistic parsimonious Turing reductions. 1. Introduction One of the most celebrated results in algebraic geometry is Weil's theorem on the number of points on algebraic curves over a finite field. In this paper, we address some computational problems related to this question. Our main results are: ffi A "computational Weil estimate" for projections of curves and images of polynomials, in Section 3. ffi #Pcompleteness of the exact counting problem for sparse curves, in Section 4. We consider a finite field F q with q elements, an algebraic closure K of F q , a polynomial f 2 F q [x; y] of degree n , the plane curve C = ff = 0...
Frobenius Pseudoprimes
 Math. Comp
"... Abstract. The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word “pseudoprime ” in them. Examples include pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes, strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and ..."
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Cited by 7 (2 self)
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Abstract. The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word “pseudoprime ” in them. Examples include pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes, strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and Perrin pseudoprimes. Though these tests represent a wealth of ideas, they exist as a hodgepodge of definitions rather than as examples of a more general theory. It is the goal of this paper to present a way of viewing many of these tests as special cases of a general principle, as well as to reformulate them in the context of finite fields. One aim of the reformulation is to enable the creation of stronger tests; another is to aid in proving results about large classes of pseudoprimes. 1.
Giuga's Conjecture on Primality
 AMER. MATH. MONTHLY
, 1996
"... G. Giuga conjectured that if an integer n satisfies n\Gamma1 P k=1 k n\Gamma1 j \Gamma1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of ..."
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G. Giuga conjectured that if an integer n satisfies n\Gamma1 P k=1 k n\Gamma1 j \Gamma1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisfies (p \Gamma 1) j (n=p \Gamma 1) and p j (n=p \Gamma 1). Using this characterization, he proved computationally that any counterexample has at least 1,000 digits; equipped with more computing power, E. Bedocchi later raised this bound to 1,700 digits. By improving on their method, we determine that any counterexample has at least 13,800 digits. We also give some new results on the second of the above conditions. This leads, in our opinion, to some interesting questions about what we call Giuga numbers and Giuga sequences.