Showing papers on "Prime (order theory) published in 2010"

High-speed software implementation of the optimal ate pairing over Barreto-Naehrig curves

Abstract: This paper describes the design of a fast software library for the computation of the optimal ate pairing on a Barreto-Naehrig elliptic curve. Our library is able to compute the optimal ate pairing over a 254-bit prime field Fp, in just 2.33 million of clock cycles on a single core of an Intel Core i7 2.8GHz processor, which implies that the pairing computation takes 0.832msec. We are able to achieve this performance by a careful implementation of the base field arithmetic through the usage of the customary Montgomery multiplier for prime fields. The prime field is constructed via the Barreto-Naehrig polynomial parametrization of the prime p given as, p = 36t4 + 36t3 + 24t2 + 6t + 1, with t = 262 - 254 + 244. This selection of t allows us to obtain important savings for both the Miller loop as well as the final exponentiation steps of the optimal ate pairing.

The existence of two closed geodesics on every Finsler 2-sphere

Abstract: In this paper, we prove that for every Finsler metric on S 2 there exist at least two distinct prime closed geodesics.

Mining process models with prime invisible tasks

Abstract: Process mining is helpful for deploying new business processes as well as auditing, analyzing and improving the already enacted ones. Most of the existing process mining algorithms have some problems in dealing with invisible tasks, i.e., such tasks that exist in a process model but not in its event log. In this paper, a new process mining algorithm named @a^# is proposed, which extends the mining capability of the classical @a algorithm by supporting the detection of prime invisible tasks from event logs. Prime invisible tasks are divided into five types according to their structural features, i.e., INITIALIZE, SKIP, REDO, SWITCH and FINALIZE. After that, a new ordering relation for detecting mendacious dependencies between tasks that reflects prime invisible tasks is introduced. A reduction rule for identifying redundant ''mendacious'' dependencies is also considered. The construction algorithm to insert prime invisible tasks of SKIP/REDO/SWITCH types is presented. The @a^# algorithm has been evaluated using both artificial and real-life logs and the results are promising.

On Exponential Sums in Finite Fields

Abstract: The purpose of this paper is to establish certain multilinear exponential sums in arbitrary finite fields, extending some of the results from [1] for prime fields.

A Sticky Stick? The Locus of Morphological Representation in the Lexicon.

Abstract: It is demonstrated that the meaning given to an ambiguous word (e.g., stick) can be biased by the masked presentation of a polymorphemic word derived from that meaning (e.g., sticky). No bias in interpretation is observed when the masked prime is a word that is semantically related to the target with no morphological relationship (e.g., glue), though such a semantically based bias is revealed when the prime is unmasked. Because the masked priming results cannot be explained in terms of facilitation of processing at either the form level or the semantic level, it is concluded that an intermediate level provides the locus of the effect, referred to here as the ‘lemma’ level. Thus, any model of lexical processing needs to incorporate such an intermediate level to capture the relationship between stem morphemes and their derived forms.

New Constructions of Binary Sequences With Optimal Autocorrelation Value/Magnitude

Abstract: In this paper, we give three new constructions of binary sequences of period AN with optimal autocorrelation value or optimal autocorrelation magnitude using N × 4 interleaved sequences. Yu and Gong recently found any binary sequence of period AN with optimal autocorrelation value constructed from an almost difference set by Arasu et al. is an N × 4 interleaved sequence for which all four columns in its N × 4 array are shift equivalent up to the complement. We found that it is not necessary that four columns are shift equivalent. Instead, it could be a pair of related sequences together with their shifts as the column sequences. The first construction is to use a generalized GMW sequence of period N = 2k - 1 and its modified version, the second construction is to use a twin prime sequence of length N = p(p + 2) and its modified version, and the third construction, a pair of Legendre sequences of period N = p (p odd prime) with their respective first terms complementary (the 2-level autocorrelation property is not needed for the Legendre sequence). The comparison with the known constructions are given. For the new sequences with optimal autocorrelation value, their corresponding new almost difference sets are also derived.

Congruences concerning Legendre polynomials

Abstract: Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

Optimal Frequency-Hopping Sequences With New Parameters

Abstract: A frequency-hopping sequence (FHS) of length v and frequency set size M is called a (v,M,?)-FHS if its maximum out-of-phase Hamming autocorrelation is ?. Three new classes of optimal FHSs with respect to the Lempel-Greenberger bound are presented in this paper. First, new optimal (p,M,f)-FHSs are constructed when p = Mf +1 is an odd prime such that f is even and p ? 3 mod 4 . And then, a construction for optimal (kp,p,k)-FHSs is given for any odd prime p and a positive integer K < p such that k = 2,4,p1,p1(p1 + 2 ),2m -1,or p1 m-1, where p1 and p1+2 are odd primes. Finally, several new optimal FHSs with maximum out-of-phase Hamming autocorrelation 1 or 2 are also presented. In particular, the existence of optimal (v,N,1)-FHSs is proven for any integer N ? 3 and any integer v with N +1 ? v ? 2 N-1, as well as the existence of optimal (2N +1,N,2)-FHSs is shown for any integer N ? 3. These classes of optimal FHSs have new parameters which are not covered in the literature.

A note on lattices in semi-stable representations

Abstract: Let p be a prime, K a finite extension over \({{\mathbb Q}_p}\) and \({G = {\rm Gal}(\overline K /K)}\) . We extend Kisin’s theory on \({\varphi}\) -modules of finite E(u)-height to give a new classification of G-stable \({{\mathbb Z}1_p}\) -lattices in semi-stable representations.

On the probabilities of local behaviors in abelian field extensions

Abstract: For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

The Conjectures of Alon-Tarsi and Rota in Dimension Prime Minus One

Abstract: A formula for Glynn's hyperdeterminant $\det_p$ ($p$ prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums $p-1$ into $p-1$ permutation matrices with even product, minus the number of ways with odd product, is 1 (mod $p$). It follows that the number of even Latin squares of order $p-1$ is not equal to the number of odd Latin squares of that order. Thus Rota's basis conjecture is true for a vector space of dimension $p-1$ over any field of characteristic zero or $p$, and all other characteristics except possibly a finite number. It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota's basis conjecture for vector spaces of even dimension over a field is unsolved.

Further progress on difference families with block size 4 or 5

Abstract: A strong indication about the existence of a (7p, 4, 1) difference family with p ? 7 (mod 12) a prime has been given in [11]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq, 4, 1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq, 5, 1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq, 4, 1) and (pq, 5, 1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ? 11 (mod 20) not exceeding 10,000.

THE DIOPHANTINE EQUATION A4 + 2δB2 = Cn

Abstract: In a previous paper, the second author proved that the equation had no integral solutions for prime p > 211 and (A,B,C) = 1. In the present paper, we explain how to extend this result to smaller exponents, and to the related equation

Limitations on Transformations from Composite-Order to Prime-Order Groups: The Case of Round-Optimal Blind Signatures

Abstract: Beginning with the work of Groth and Sahai, there has been much interest in transforming pairing-based schemes in composite-order groups to equivalent ones in prime-order groups. A method for achieving such transformations has recently been proposed by Freeman, who identified two properties of pairings using composite-order groups—“cancelling” and “projecting”—on which many schemes rely, and showed how either of these properties can be obtained using prime-order groups.

PRIME Interoperability Tests and Results from Field

Abstract: PRIME (PoweRline Intelligent Metering Evolution) is one of the prominent upcoming powerline communication technologies, targeted for use in smart metering applications. The PRIME PHY / MAC specifications are open, publicly available and are developed by the PRIME Alliance, an industry consortium that includes utilities, meter vendors and semiconductor suppliers. PRIME employs OFDM modulation in the CENELEC A band (9 - 95 kHz), and achieves data rates from 21 kbps to 128 kbps at the PHY layer. The PRIME MAC is optimized for tree- topology networks, and features a novel node discovery and network building process. PRIME converges to IPv4 and IEC 61334-4-32 at the network layer, and is evolving to support IPv6. In this paper, we describe the PRIME Alliance, and review technical details of the PRIME PHY and MAC. We review the certification and interoperability tests defined by the PRIME Alliance, to ensure openness and future-proof technical performance with multi-vendor solutions. We present some initial results from small-scale PRIME field deployments.

Complex equiangular Parseval frames and Seidel matrices containing $p$th roots of unity

Abstract: We derive necessary conditions for the existence of complex Seidel matrices containing pth roots of unity and having exactly two eigenvalues, under the assumption that p is prime. The existence of such matrices is equivalent to the existence of equiangular Parseval frames with Gram matrices whose off-diagonal entries are a common multiple of the pth roots of unity. Explicitly examining the necessary conditions for p = 5 and p = 7 rules out the existence of many such frames with a number of vectors less than 50, similar to previous results in the cube roots case. On the other hand, we confirm the existence of p 2 × p 2 Seidel matrices containing pth roots of unity, and thus the existence of the associated complex equiangular Parseval frames, for any p > 2. The construction of these Seidel matrices also yields a family of previously unknown Butson-type complex Hadamard matrices.

Prime and almost prime integral points on principal homogeneous spaces

Abstract: We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.

Congruences between Hilbert modular forms: constructing ordinary lifts

Abstract: In this note we improve on the results of our earlier paper[BLGG12], proving a near-optimal theorem on the existence of ordinary lifts of a mod l Hilbert modular form for any odd prime l.

Prime regular Hopf algebras of GK-dimension one

Abstract: This paper constitutes the first part of a program to classify all affine prime regular Hopf algebras H of Gelfand-Kirillov dimension one over an algebraically closed field of characteristic zero. We prove a number of properties of such an algebra, list some classes of examples and then prove that, when the PI-degree of H is prime, our list contains all such algebras.

A new look at the Jordan-Hölder theorem for semimodular lattices

Abstract: We show that in a semimodular lattice L of finite length, from any prime interval we can reach any maximal chain C by an up- and a down-perspectivity. Therefore, C is a congruence-determining sublattice of L.

Congruences concerning Legendre polynomials

Abstract: Let p be an odd prime. In this paper, by using the properties of Legendre polynomials we prove some congruences for Σ p―1 2 k=0 ( 2k k m ―k (mod p 2 ). In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.

On monochromatic solutions of some nonlinear equations in ℤ/pℤ

Abstract: Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ℤ such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ℤ/pℤ, where p is a prime, and obtain an even stronger density result.