Showing papers on "Prime (order theory) published in 2000"

Selecting Cryptographic Key Sizes.

Abstract: In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems.

Distribution of the partition function modulo $m$

Abstract: Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).

New bounds for Gauss sums derived from KTH powers, and for Heilbronn's exponential sum

Abstract: where p is prime, e(x) = exp(2πix), and ep(x) = e(x/p). In each case we shall assume that p | / a unless the contrary is explicitly stated. Gauss sums arise in investigations into Waring’s problem, and other additive problems involving k-th powers. Although they are amongst the simplest complete exponential sums, the question as to their true order of magnitude is far from being resolved. We remark at the outset that if (k, p− 1) = k0, then Gp(a, k) = Gp(a, k0). Thus it suffices to suppose, as indeed we shall, that k|p− 1. When p | / a the trivial bound for G(a) states that |G(a)| ≤ p. The next simplest estimate takes the form |G(a)| ≤ (k − 1)√p. (1) This may be obtained by writing G(a) in terms of the character Gauss sum as

A Gaussian hypergeometric series evaluation and Apéry number congruences

Abstract: If p is prime, then let φp denote the Legendre symbol modulo p and let p be the trivial character modulo p. As usual, let n+1Fn(x)p := n+1Fn „ φp, φp, . . . , φp p, . . . , p | x « p be the Gaussian hypergeometric series over Fp. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain. Here we take the first step by obtaining a simple formula for 4F3(1)p. As a corollary we obtain a result describing the distribution of traces of Frobenius for certain families of elliptic curves. We also find that 4F3(1)p satisfies surprising congruences modulo 32 and 11. We then establish a mod p2 “supercongruence” between Apery numbers and the coefficients of a certain eta-product; this relationship was conjectured by Beukers in 1987. Finally, we obtain many new mod p congruences for generalized Apery numbers.

Counting Points on Hyperelliptic Curves over Finite Fields

Abstract: We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm a la Schoof for genus 2 using Cantor's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature.

A Signcryption Scheme Based on Integer Factorization

Abstract: Signcryption is a public-key cryptographic primitive introduced by Zheng, which achieves both message confidentiality and nonrepudiatable origin authenticity, at a lower computational and communication overhead cost than the conventional 'sign-then-encrypt' approach. We propose a new signcryption scheme which gives a partial solution to an open problem posed by Zheng, namely to find a signcryption scheme based on the integer factorization problem. In particular, we prove that our scheme is existentially unforgeable, in the random oracle model, subject to the assumption that factoring an RSA modulus N = pq (with p and q prime) is hard even when given the additional pair (g; S), where g ∈ Z*N is an asymmetric basis of large order less than a bound S=2 « √N.

Emergence of Features in Metaphor Comprehension

Abstract: The interaction theory of metaphor holds that a metaphor can cause emergence of features that are not associated with the topic or vehicle of the metaphor. We present 2 studies on emergent features. In the 1st study, using a production task, we found that over 60% of the features produced during metaphor processing are emergent features. We also found that giving explicit instructions to the participants to use imagery does not have any significant impact on the number of emergent features. In the 2nd experiment, we used response time as an indicator of the activation levels of emergent features. The emergent features require a longer response time than topic-only and vehicle-only features, when tested with topic or vehicle primes. When tested with the full metaphor as the prime, the topic-only and vehicle-only features gave long response times. But the emergent features did not change their duration from one prime condition to another. We suggest these results are consistent with the interaction theory o...

Social learning and costly information acquisition

Abstract: Short-lived agents want to predict a random variable $\theta $ and have to decide how much effort to devote to collect private information and consequently how much to rely on public information. The latter is just a noisy average of past predictions. It is shown that costly information acquisition prevents an unbounded accumulation of public information if (and only if) the marginal cost to acquire information is positive at zero $(C^\prime (0) > 0)$ . When $C^\prime (0) = 0$ public precision at period n, $\tau_n$ , tends to infinity with n but the rate of convergence of public information to $\theta $ is slowed down with respect to the exogenous information case. At the market outcome agents acquire too little private information. This happens either with respect to a (decentralized) first best benchmark or, for n large, with respect to a (decentralized) second best benchmark. For high discount factors the limit point of market public precision always falls short of the welfare benchmarks whenever $C^\prime (0) > 0$ . In the extreme, as the discount factor tends to one public precision tends to infinity in the welfare-optimal programs while it remains bounded at the market solution. Otherwise, if $C^\prime (0) = 0$ public precision accumulates in an unbounded way both at the first and second best solutions. More public information may hurt at either the market or second best solutions.

Efficient Generation of Prime Numbers

Abstract: The generation of prime numbers underlies the use of most public-key schemes, essentially as a major primitive needed for the creation of key pairs or as a computation stage appearing during various cryptographic setups. Surprisingly, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptographic usages, prime number generation algorithms remain scarcely investigated and most real-life implementations are of rather poor performance. Common generators typically output a n-bit prime in heuristic average complexity O(n4) or O(n4/ log n) and these figures, according to experience, seem impossible to improve significantly: this paper rather shows a simple way to substantially reduce the value of hidden constants to provide much more efficient prime generation algorithms. We apply our techniques to various contexts (DSA primes, safe primes, ANSI X9.31-compliant primes, strong primes, etc.) and show how to build fast implementations on appropriately equipped smart-cards, thus allowing on-board key generation.

A fast algorithm for determining the linear complexity of a sequence with period p/sup n/ over GF(q)

Abstract: A fast algorithm is presented for determining the linear complexity of a sequence with period p/sup n/ over GF (q), where p is an odd prime, and where q is a prime and a primitive root (mod p/sup 2/).

Blowing Ups of 3-dimensional Terminal singularities, II

Abstract: We study blowing ups of 3-dimensional terminal singularities of type (cD/2) such that the exceptional loci are prime divisors and have discrepancies 1/2. We determined such blowing ups completely.

δ-derivations of prime alternative and mal’tsev algebras

Abstract: We give a description of δ-derivations of prime alternative and non-Lie Mal’tsev Φ-algebras, with some restrictions on an operator ring Φ. For algebras in these classes, every δ-derivation is proved trivial.

Young Children, Minimal Groups, and Dichotomous Categorization

Abstract: Research suggests that ingroup bias in the minimal group paradigm may rely on dichotomous categorization, not social categorization per se. Dichotomous categorization may prime competition because of its unique cultural significance. Young children often do not demonstrate the culturally shaped cognitive tendencies of their elders, even though they can. Thus, young children may not show bias in the minimal group context. Two experiments examined these issues. In Experiment 1, children completed a minimal groups task in two-or three-group conditions. They received no prime, a neutral prime, or a competitive prime. As predicted, children did not display ingroup bias in two-or three-group conditions unless competitively primed. In Experiment 2, undergraduate students completed a minimal groups task in two-or three-group conditions. They received no prime or a competitive prime. As predicted, undergraduates displayed bias in two-group contexts. They displayed bias in three-group contexts only when competitive...

On the locus of the semantic satiation effect: Evidence from event-related brain potentials

Abstract: The present study sought to determine whether semantic satiation is merely a by-product of adaptation or satiation of upstream, nonsemantic perceptual processes or whether the effect can have a locus in semantic memory. This was done by measuring event-related brain potentials (ERPs) in a semantic word-detection task involving multiple presentations of primes and critical related and unrelated words in three experiments involving visual (Experiment 1) and auditory (Experiments 2A and 2B) stimuli. Primes varied in their type case (Experiment 1) or pitch (Experiment 2B) in order to discourage sensory adaptation. Prime satiation and relatedness of the primes to the critical word had interacting effects on ERP amplitude to critical words, particularly within the time-window of the N400 component. Because numerous studies have indicated a role for the N400 in semantic processing, modulation of the N400 relatedness effect by prime satiation (with little or no contribution from perceptual adaptation) suggests that semantic memory can be directly satiated, rather than the cost to semantic processing necessarily resulting from impoverishment of perceptual inputs.

Finite quantum groups over abelian groups of prime exponent

Abstract: Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new examples. We show that any finite dimensional pointed Hopf algebra with group of prime exponent (greater than 17) is indeed in this family. An important step in the proof follows from a another result, where we show that a wide family of finite dimensional pointed Hopf algebras is generated by group-like and skew-primitive elements, giving additional support to a conjecture in our previous paper "Finite quantum groups and Cartan matrices", Adv. Math. 154 (2000), 1--45.

On Vorontsov’s Theorem on K3 surfaces with non-symplectic group actions

Abstract: We shall give a proof for Vorontsov’s Theorem and apply this to classify log Enriques surfaces with large prime canonical index.

Prime Movers: Define Your Business or Have Someone Define it Against You

Abstract: The New Logic of Value Creation. Configuration and Reconfiguration. The Perception of Value. An Emerging Architecture for Value Creation. Customer Orientation: Aligning the Value--Creation Logics of the Firm with the Value--Creation Logics of Existing Customers. Capability Focus and Lean Management: Leveraging the Existing Capabilities of the Firm to Create New Offerings. Actor--Centered Market Making: Leveraging New Capabilities of the Firm to Create New Offerings. Co--operative Market Making: Prime Movers Acting for Others, Not Themselves. Appreciative Market Making: When Becoming a Prime Mover is Not the Driving Goal. Value Creation as Prime Movers See It. Notes. References and Further Reading. Index.

Classification of semisimple Hopf algebras of dimension 16

Abstract: In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.

My numbers, my friends : popular lectures on number theory

Abstract: The Fibonacci Numbers and the Arctic Ocean.- Representation of Real Numbers by Means of Fibonacci Numbers.- Prime Number Records.- Selling Primes.- Euler's Famous Prime Generating Polynomial and the Class Number of Imaginary Quadratic Fields.- Gauss and the Class Number Problem.- Consecutive Powers.- 1093.- Powerless Facing Powers.- What Kind of Number Is $$ \sqrt 2 ^{\sqrt 2 } $$ ?.- Galimatias Arithmeticae.

On the flatness of local models for the symplectic group

Abstract: We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.

The exceptional set in the four prime squares problem

Abstract: In this paper we prove that, with at most $O(N^{13/15+\varepsilon})$ exceptions, all positive even integers $n \leq N$ with $n \equiv 4 (\mathrm{mod}\,24)$ can be written as sums of four squares of primes.

On Lie ideals and Jordan left derivations of prime rings

Abstract: Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^{2} \in U$ for all $u \in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d(u^{2})=2ud(u)$ for all $u \in U$, then $d(uv)=ud(v)+vd(u)$ for all $u,v \in U$.

Low Rank Co-Diagonal Matrices and Ramsey Graphs

Abstract: We examine nn matrices overZm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1=(p 1) O(1) ). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c p logn log logn). We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs.

On integers not of the form +/-p^a +/-q^b

Abstract: In 1975 F. Cohen and J.L. Selfridge found a 94-digit positive integer which cannot be written as the sum or dierence of two prime powers. Following their basic construction and introducing a new method to avoid a bunch of extra congruences, we are able to prove that if

Recognition of Alternating Groups of Degrees r+1 and r+2 for Prime r and the Group of Degree 16 by Their Element Order Sets

Abstract: It is proved that a finite group whose element order set is the same as that of an alternating group An of degree n=r+1 or r+2 for prime r>5 or n=16 is isomorphic to An.

Note on mod p Siegel modular forms II

Abstract: The structure of the ring of mod p Siegel modular forms of degree two is determined in the cases where the prime p is 2 or 3.