Showing papers on "Prime (order theory) published in 1997"

Lower bounds for discrete logarithms and related problems

Abstract: This paper considers the computational complexity of the discrete logarithm and related problems in the context of "generic algorithms"--that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as a unique binary string. Lower bounds on the complexity of these problems are proved that match the known upper bounds: any generic algorithm must perform Ω(p1/2) group operations, where p is the largest prime dividing the order of the group. Also, a new method for correcting a faulty Diffie-Hellman oracle is presented.

Some optimal inapproximability results

Abstract: We prove optimal, up to an arbitrary †>0, inapproximability results for Max-Ek-Sat for k‚ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover. Categories and Subject Descriptors: F2.2 (Analysis of Algorithms and Problem Complexity): Non- numerical Algorithms and Problems

P-adic Banach spaces and families of modular forms

Abstract: Let p be a prime, Cp the completion of an algebraic closure of the p-adic numbers Qp and K a nite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x| = p−v(x) for x ∈ Cp. Suppose N is a positive integer prime to p. Let X1(Np) denote the modular curve over K which represents elliptic curves with 1(Np)-structure and let Up be the Hecke operator on modular forms on X1(Np) which takes a form with q-expansion ∑ n anq n to the modular form with q-expansion ∑ n anpq n. A modular form F is said to have slope ∈ Q if there is a polynomial R(T ) over Cp such that R(Up)F = 0 and such that the Newton polygon of R(T ) has only one side and its slope is − .

Social Impact Research: Personal Computers, Mass Media, and Use of Time

Abstract: One measure of the impact of the personal computer is in terms of its time displacement of other activities. Mass media generally are considered a prime candidate given the on-line profusion of inf...

A rigid analytic Gross-Zagier formula and arithmetic applications

Abstract: Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.

On the p -Divisibility of Fermat quotients

Abstract: The authors carried out a numerical search for Fermat quotients Q a = (a p-1 - 1)/p vanishing mod p, for 1 ≤ a ≤ p - 1, up top < 10 6 . This article reports on the results and surveys the associated theoretical properties of Q a . The approach of fixing the prime p rather than the base a leads to some aspects of the theory apparently not published before.

Expression of the C4 Me1 Gene from Flaveria bidentis Requires an Interaction between 5[prime] and 3[prime] Sequences.

Abstract: The efficient functioning of C4 photosynthesis requires the strict compartmentation of a suite of enzymes in either mesophyll or bundle sheath cells. To determine the mechanism controlling bundle sheath cell-specific expression of the NADP-malic enzyme, we made a set of chimeric constructs using the 5[prime] and 3[prime] regions of the Flaveria bidentis Me1 gene fused to the [beta]-glucuronidase gusA reporter gene. The pattern of GUS activity in stably transformed F. bidentis plants was analyzed by histochemical and cell separation techniques. We conclude that the 5[prime] region of Me1 determines bundle sheath specificity, whereas the 3[prime] region contains an apparent enhancer-like element that confers high-level expression in leaves. The interaction of 5[prime] and 3[prime] sequences was dependent on factors that are present in the C4 plant but not found in tobacco.

On additive maps of prime rings satisfying the engel condition

Abstract: Let A be a prime ring with nonzero right ideal R and f : R → A an additive map. Next, let k,n1, n2,…,nk be natural numbers. Suppose that […[[(x), xn1], xn2],…, xnk]=0 for all x ∈ R. Then it is proved in Theorem 1.1 that [f(x),x]=0 provided that either char(A)=0 or char (A)> n1+n2+ …+nk Theorem 1.1 is a simultaneous generalization of a number of results proved earlier.

Zero-Knowledge Proofs for Finite Field Arithmetic or: Can Zero-Knowledge be for Free?

Abstract: We present zero-knowledge proofs and arguments for arithmetic circuits over finite prime fields, namely given a circuit, show in zero-knowledge that inputs can be selected leading to a given output. For a field GF(q), where q is an n-bit prime, a circuit of size O(n), and error probability 2^−n, our protocols require communication of O(n^2) bits. This is the same worst-cast complexity as the trivial (non zero-knowledge) interactive proof where the prover just reveals the input values. If the circuit involves n multiplications, the best previously known methods would in general require communication of Omega(n^3 log n) bits. Variations of the technique behind these protocols lead to other interesting applications. We first look at the Boolean Circuit Satisfiability problem and give zero-knowledge proofs and arguments for a circuit of size n and error probability 2^−n in which there is an interactive preprocessing phase requiring communication of O(n^2) bits. In this phase, the statement to be proved later need not be known. Later the prover can non-interactively prove any circuit he wants, i.e. by sending only one message, of size O(n) bits. As a second application, we show that Shamirs (Shens) interactive proof system for the (IP-complete) QBF problem can be transformed to a zero-knowledge proof system with the same asymptotic communication complexity and number of rounds. The security of our protocols can be based on any one-way group homomorphism with a particular set of properties. We give examples of special assumptions sufficient for this, including: the RSA assumption, hardness of discrete log in a prime order group, and polynomial security of Die-Hellman encryption. We note that the constants involved in our asymptotic complexities are small enough for our protocols to be practical with realistic choices of parameters.

CNF and DNF Considered Harmful for Computing Prime Implicants/Implicates

Abstract: Several methods to compute the prime implicants and the prime implicates of a negation normal form (NNF) formula are developed and implemented. An algorithm PI is introduced that is an extension to negation normal form of an algorithm given by Jackson and Pais. A correctness proof of the PI algorithm is given. The PI algorithm alone is sufficient in a computational sense. However, it can be combined with path dissolution, and it is shown empirically that this is often an advantage. None of these variations rely on conjunctive normal form or on disjunctive normal form. A class of formulas is described for which reliance on CNF or on DNF results in an exponential increase in the time required to compute prime implicants/implicates. The possibility of avoiding this problem with efficient structure preserving clause form translations is examined briefly and appears unfavorable.

Random Combinatorial Structures and Prime Factorizations

Abstract: Techniques for counting salient features of randomly selected elements of very large sets, with discussion in the particular context of the factorization of large integers.

John Philoponus' New Definition of Prime Matter: Aspects of Its Background in Neoplatonism and the Ancient Commentary Tradition

Abstract: This is the first full discussion of Philoponus' account of matter. It is shown here that philosophical problems in Neoplatonism motivated the definition of prime matter as three-dimensional extension, and that Plotinus, Syrianus, and Proclus prepared the way for Philoponus.

On a problem of Byrnes concerning polynomials with restricted coefficients

Abstract: As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length N*(m) of a polynomial with all coefficients in {-1, 1} which has a zero of a given order m at x = 1. In that paper we showed that N*(m) = 2 m for all m < 5 and showed that the extrernal polynomials for were those conjectured by Byrnes, but for m = 6 that N*(6) = 48 rather than 64. A polynomial with N = 48 was exhibited for m = 6, but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that N*(7) is one of the 7 values 48, 56, 64, 72, 80, 88 or 96. Here we prove that N* (7) = 96 without determining all extremal polynomials. We also make some progress toward determining N*(8). As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if ζ p is a primitive pth root of unity where p < m + 1 is a prime, then the condition that all coefficients of P be in {-1, 1}, together with the requirement that P(x) be divisible by (x - 1) m puts severe restrictions on the possible values for the cyclotomic integer P(ζ p ).

Modular representation of finite groups of Lie type in non-defining characteristic

Abstract: Let us consider a connected reductive algebraic group G, defined over the finite field F q with corresponding Frobenius morphism F. We are concerned here with properties of finite-dimensional modules for the finite group G F over a sufficiently large field k of characteristic l where l is a prime not dividing q.

Three levels of elections in Israel: The 1996 party, parliamentary and prime ministerial elections

Abstract: This article explains the methods used to select parliamentary candidates for Israeli's national list system of PR and assesses the effects on parliament and the political parties of electing the prime minister in a separate ballot.

Groups with few non-nilpotent subgroups

Abstract: Let G be a non-nilpotent group in which all proper subgroups are nilpotent. If G is finite then G is soluble [18], and a classification of such groups is given in [14]. The paper [12]. of Newman and Wiegold discusses infinite groups with this property. Clearly such a group is either finitely generated or locally nilpotent. Many interesting results concerning the finitely generated case are established in [12]. Since the publication of that paper there have appeared the examples due to Ol'shanskii and Rips (see [13]) of finitely generated infinite simple p-groups all of whose proper nontrivial subgroups have order p, a prime. Following [12], let us say that a group G is an AN-group if it is locally nilpotent and non-nilpotent with all proper subgroups nilpotent. A complete description is given in Section 4 of [12] of AN-groups having maximal subgroups. Every soluble AN-gvoup has locally cyclic derived factor group and is a p-group for some prime p ([12; Lemma 4.2]). The only further information provided in [12] on AN-groups without maximal subgroups is that they are countable. Four years or so after the publication of [12], there appeared the examples of Heineken and Mohamed [5]: for every prime p there exists a metabelian, non-nilpotent p-group G having all proper subgroups nilpotent and subnormal; further, G has no maximal subgroups and so G/G' is a Prufer p-group in each case.

Bounds for fixed point free elements in a transitive group and applications to curves over finite fields

Abstract: We investigateV f , the cardinality of the value set of a polynomialf of degreen over a finite field of cardinalityq It has been shown that iff is not bijective, thenV f ≤q−(q−1)/n Polynomials do exist which essentially achieve that bound We do prove that if the degree off is prime to the characteristic andf is not bijective, then asymptoticallyV f ≤(5/6)q We consider related problems for curves and higher dimensional varieties This problem is related to the number of fixed point free elements in finite groups, and we prove some results in that setting as well

Comments on search procedures for primitive roots

Abstract: Let p be an odd prime. Assuming the Extended Riemann Hypothesis, we show how to construct O((logp) 4 (loglogp) -3 ) residues modulo p, one of which must be a primitive root, in deterministic polynomial time. Granting some well-known character sum bounds, the proof is elementary, leading to an explicit algorithm.

Inclusive $(e,e^\prime N)$, $(e,e^\prime NN)$, $(e,e^\prime \pi)$ ... reactions in nuclei

Abstract: We study the inclusive $(e,e^\prime N)$, $(e,e^\prime NN)$, $(e,e^\prime \pi)$, $(e,e^\prime \pi N)$ reactions in nuclei using a Monte Carlo simulation method to treat the multichannel problem of the final state. The input consists of reaction probabilities for the different steps evaluated using microscopical many body methods. We obtain a good agreement with experiment in some channels where there is data and make predictions for other channels which are presently under investigation in several electron laboratories. The comparison of the theoretical results with experiment for several kinematical conditions and diverse channels can serve to learn about different physical processes ocurring in the reaction. The potential of this theoretical tool to make prospections for possible experiments, aiming at pinning down certain reaction probabilities, is also emphasized.

The least prime primitive root and the shifted sieve

Abstract: We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was log p to an exponent tending to infinity with p, lies in the use of the linear sieve (a particular version called the shifted sieve) rather than Brun's sieve. The same methods allow us to rederive a conditional result of Shoup on the least prime primitive root (mod p) for all prime moduli p, assuming the generalized Riemann hypothesis. We also extend both results to composite moduli q, where the analogue of a primitive root is an element of maximal multiplicative order (mod q).

Model-Checking for a Subclass of Event Structures

Abstract: A finite representation of the prime event structure corresponding to the behaviour of a program is suggested. The algorithm of linear complexity using this representation for model checking of the formulas of Discrete Event Structure Logic without past modalities is given. A method of building finite representations of event structures in an efficient way by applying partial order reductions is provided.

Globally Irreducible Representations of Finite Groups and Integral Lattices

Abstract: The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L 2(q) and 2B2(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pn-1)/2 of the symplectic group Sp2n(p) (p ≡ 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pn−1). A summary of currently known globally irreducible representations is given.