Topic
Divisor
About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this article, it was shown that P (∆(1, k)) ≤ k and therefore, the assumption n > k in (1) cannot be removed, and that the assumption in (2) cannot even be removed if n is greater than or equal to the least prime exceeding 2k.
Abstract: We observe that P (∆(1, k)) ≤ k and therefore, the assumption n > k in (1) cannot be removed. For n > k, Moser [5] sharpened (1) to P (∆(n, k)) > 11 10 k and Hanson [3] to P (∆(n, k)) > 1.5k unless (n, k) = (3, 2), (8, 2), (6, 5). Further Faulkner [2] proved that P (∆(n, k)) > 2k if n is greater than or equal to the least prime exceeding 2k and (n, k) 6= (8, 2), (8, 3). In this paper, we sharpen the results of Hanson and Faulkner. We shall not use these results in the proofs of our improvements. We prove
17 citations
TL;DR: In this article, the smallest known complete arcs in affine spaces of dimension n ≥ 4 were constructed from singular cubic curves with a characteristic greater than three points, where n ≥ 1 4 q 1/4.
Abstract: Small complete arcs and caps in Galois spaces over finite fields $$\mathbb {F}_q$$ F q with characteristic greater than three are constructed from singular cubic curves. For $$m$$ m a divisor of $$q+1$$ q + 1 or $$q-1$$ q - 1 , complete plane arcs of size approximately $$q/m$$ q / m are obtained, provided that $$(m,6)=1$$ ( m , 6 ) = 1 and $$m<\frac{1}{4}q^{1/4}$$ m < 1 4 q 1 / 4 . If in addition $$m=m_1m_2$$ m = m 1 m 2 with $$(m_1,m_2)=1$$ ( m 1 , m 2 ) = 1 , then complete caps in affine spaces of dimension $$N\equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) with roughly $$\frac{m_1+m_2}{m}q^{N/2}$$ m 1 + m 2 m q N / 2 points are described. These results substantially widen the spectrum of $$q$$ q s for which complete arcs in $$AG(2,q)$$ A G ( 2 , q ) of size approximately $$q^{3/4}$$ q 3 / 4 can be constructed. Complete caps in $$AG(N,q)$$ A G ( N , q ) with roughly $$q^{(4N-1)/8}$$ q ( 4 N - 1 ) / 8 points are also provided. For infinitely many $$q$$ q s, these caps are the smallest known complete caps in $$AG(N,q)$$ A G ( N , q ) , $$N \equiv 0 \pmod 4$$ N ? 0 ( mod 4 ) .
17 citations
TL;DR: If N is a bicomposite RSA modulus, the analysis shows how to quantify the risk posed by an iterated encryption attack using the prime generation algorithm defined in Maurer maur.
Abstract: Let N and a be integers larger than 1 Define an orbit to be the collection of residues in _N^* generated by iteratively applying x → x ^α mod N to an element x ∈ Z_N^* which eventually maps back to itself An orbit‘s length is the number of distinct residues in the orbit When N is a large bicomposite integer, such as is commonly used in many cryptographic applications, and when certain prime factorizations related to N are known, all orbit lengths and the number of orbits of each possible length can be efficiently computed using the results presented If the required integer factorizations are only partially known, the risk that a randomly selected periodic element might produce an orbit shorter than some (typically large) divisor of (p(N)) can be bounded The information needed to produce such a bound is fully available when the prime factors of N are generated using the prime generation algorithm defined in Maurer maur Results presented can assist in choosing wisely a modulus N for the Blum, Blum, and Shub pseudo-random bit generator If N is a bicomposite RSA modulus, the analysis shows how to quantify the risk posed by an iterated encryption attack
17 citations
TL;DR: In this paper, the divisor theory of the Kontsevich moduli spaces M0,0(G(k, n), d) of genus-zero stable maps to the Grassmannians was studied.
Abstract: In this note we study the divisor theory of the Kontsevich moduli spacesM0,0(G(k, n), d) of genus-zero stable maps to the Grassmannians. We calculate the classes of several geometrically significant divisors. We prove that the cone of effective divisors stabilizes as n increases and we determine the stable effective cone. We also characterize the ample cone.
17 citations
TL;DR: The scrollar invariants of a projective non-hyperelliptic curve of genus g defined over the field C of complex numbers were defined by Ohbuchi et al..
Abstract: Let C be an irreducible smooth projective non-hyperelliptic curve of genus g defined over the field C of complex numbers. Let g\ be a complete base-point free special linear system on C. The scrollar invariants of g\ are defined as follows. Let C be canonically embedded in P~ and let X be the union of the linear spans (D) with D e g\ . This defines a set of integers e\ > ... > ek-i > 0 such that X is the image of the projective bundle P(e\\... e^-i) = P(OPι (ei) Θ . . . Θ OPι (βk-i)) using the tautological bundle (see e.g. [2]; [7]). Those integers e ι ;e 2 ; . . . e^-i are called the scrollar invariants of g\. Those scrollar invariants determine (and are determined by) the complete linear systems associated to multiples of the linear system g\. For 1 i Here KC denotes a canonical divisor on C. Let m = βfc-ι+2. Then m is defined by the following conditions: dim(|(ra—l)p£|) — m—1 and dim(|ra ra. In case \mg\\ is birationally very ample then the scrollar invariants satisfy the inequalities e* '^'~^m 1 -f (x — (j — l)m + l) j . Equality (if not in conflict with the Riemann-Roch Theorem) can be expected being the most general case for a fixed value of m. The inequalities also imply dim(|(λ; — l)mg\.\) = dim(|((fc — l)ra 1)^ | ) 4k. This implies that \((k — l)m l)g\\ is not special. Using the dimension bound one obtains g < [(k k)m 2k + 2]/2. (This easy but interesting consequence from the inequalities is not mentioned by Kato and Ohbuchi.) In this paper we prove the following theorem.
17 citations