Showing papers on "Ring of integers published in 1994"

Representing Boolean functions as polynomials modulo composite numbers

Abstract: Define the MOD m -degree of a boolean functionF to be the smallest degree of any polynomialP, over the ring of integers modulom, such that for all 0–1 assignments $$\vec x$$ , $$F(\vec x) = 0$$ iff $$P(\vec x) = 0$$ . We obtain the unexpected result that the MOD m -degree of the OR ofN variables is $$O(\sqrt[\tau ]{N})$$ , wherer is the number of distinct prime factors ofm. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple ofn and is one otherwise. We show that the MOD m -degree of both the MOD n and $$ eg MOD_n$$ functions isN Ω(1) exactly when there is a prime dividingn but notm. The MOD m -degree of the MOD m function is 1; we show that the MOD m -degree of $$ eg MOD_m$$ isN Ω(1) ifm is not a power of a prime,O(1) otherwise. A corollary is that there exists an oracle relative to which the MOD m P classes (such as ⊕P) have this structure: MOD m P is closed under complementation and union iffm is a prime power, and MOD n P is a subset of MOD m P iff all primes dividingn also dividem.

Approximating rings of integers in number fields

Abstract: In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic mimber field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for facto- ring integers. Applying a variar.t of a Standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main cor.cern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tarne" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.

Periods of Cusp forms and elliptic curves over imaginary quadratic fields

Abstract: In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups , where n is an ideal in the ring of integers R of K. This continues work of the first author and forms part of the Ph.D. thesis of the second author. In each case we compute numerically the value of the L-series at and compare with the value of which is predicted by the Birch-Swinnerton-Dyer conjecture, finding agreement to several decimal places. In particular, we find that whenever has a point of infinite order. Several examples are given in detail from the extensive tables computed by the authors.

Real Hilbertianity and the Field of Totally Real Numbers

Abstract: We use moduli spaces for covers of the Riemann sphere to solve regular embedding problems, with prescribed extendability of orderings, over PRC fields. As a corollary we show that the elementary theory of Q tr is decidable. Since the ring of integers of Q tr is undecidable, this gives a natural undecidable ring whose quotient field is decidable.

A quadratic field which is Euclidean but not norm-Euclidean

Abstract: The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of $$\mathbb{Q}\left( {\sqrt {69} } \right)$$ .

Rings, fields, the Chinese remainder theorem and an extension-Part II: applications to digital signal processing

Abstract: For pt. I, see ibid., vol. 41, no. 10, p. 641-55 (1994). In Part I of the research work, we introduced an extension to the well known Chinese remainder theorem for processing polynomials with coefficients defined over a finite integer ring. We term this extension as the American-Indian-Chinese extension of the Chinese remainder theorem. A systematic procedure for factorizing a monic polynomial into pairwise relatively prime monic factor polynomials over integer rings was presented. This factorization is based on the corresponding factor polynomials, monic and relatively prime, over the associated finite field containing prime number of elements. In this paper, we study the application of the theory developed in Part I to deriving computationally efficient algorithms for performing tasks having multilinear form. Especially, we focus on the cyclic and acyclic convolution as they are two of the most frequently occurring computationally intensive tasks in digital signal processing. >

Coded modulation based on rings of integers modulo-q. 1. Block codes

Abstract: The authors aim to present an alternative multilevel (nonbinary) coding method, based on a ring of integers modulo-q (q is a nonprime integer number) which is suitable for coded modulation schemes. Some new concepts and definitions are introduced in order to build up a theoretical framework which allows a search for good multilevel codes. Such classes of block codes are also presented together with their interesting properties. Such codes can be used not only in q-PSK modulations, but also in q-QAM modulations by making a simple adaptation.

Recognizing units in number fields

Abstract: We present a deterministic polynomial-time algorithm that decides whether a power product IlkI yn' is a unit in the ring of integers of K , where K is a number field, yi are nonzero elements of K and ni are rational integers. The main algorithm is based on the factor refinement method for ideals, which might be of independent interest.

Circular Units of Function Fields

Abstract: A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields. Let m be a positive integer and let Cm = exp(27ri/m). Let Km = Q(4m) denote the mth cyclotomic field, and K+ its maximal real subfield. The ring of integers in Km (resp. K+) is Z[Cm] (resp. Z[4m + C-1]). In [2] Sinnott showed that the index of circular units in the full group of units of Q(Cm) equals the class number of Z[4m + Cm'] multiplied by a power of 2 which depends exclusively on the number of prime factors of m. Sinnott [3] subsequently generalized this result to an arbitrary abelian field. There is a parallel setup for function fields of characteristic p . Let Fq be the finite field with q elements, let RT = Fq[T] be the ring of polynomials over Fq (with T transcendental over Fq), and let Fq(T) be the field of rational functions over Fq . To each polynomial M E RT one can associate an extension KM, called the Mth cyclotomicfunctionfield, which enjoys properties analogous to those of the cyclotomic number field Km. In particular, Galovich and Rosen [1] proved the analogue of Sinnott's theorem in this setting. The purpose of this paper is to extend this unit index-class number formula to an arbitrary subfield of KM. Let k be any subfield of the Mth cyclotomic function field (M monic), G the Galois group of k over IFq(T), k+ the maximal subfield of k in which oo splits, Ok(Ok+) the integral closure of Fq[T] in k(k'), and Ok* the unit group of Ok. In ?3, we define a subgroup C of Ok*, which we call the circular units of k. Our main result is that C has finite index in Ok* and that this index may be written in the form [0k :C]=h(Ok+).c,k+ where h (Ok+) is the class number of Ok+, and ck+ is a rational number whose definition does not involve h(Ok+). We now briefly describe the contents of the rest of this paper. In ? 1, we present the relevant definitions and facts in the function field setting. We also state the analytic class number formula. In ?2, we review ordinary distributions on lFq(T)/RT, discuss an index notation, and obtain a preliminary result on the structure of a certain module. The circular units are introduced in ?3 and Received by the editors February 1, 1990 and, in revised form, November 27, 1991. 1991 Mathematics Subject Classification. Primary 1 lR58. (D 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

Faltings modular height and self-intersection of dualizing sheaf

Abstract: Let K be a number field, O_K the ring of integers of K and X a stable curve over O_K of genus g >= 2. In this note, we will prove a strict inequality ( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)), where $K_{X/S}$ is the canonically metrized dualizing sheaf of X over S = Spec(O_K) and Height_{Fal}(J(X_K)) is the Faltings modular height of the Jacobian of X_K. As corollary, for any constant A, the set of all stable curves X over O_K with ( (K_{X/S})^2 / [K : Q] ) <= A is finite under the following equivalence. For stable curves X and Y, X is equivalent to Y if X is isomorphic to Y over O_{K'} for some finite extension field K' of K.

On Code Linearity and Rotational Invariance for a Class of Trellis Codes for M-PSK

Abstract: A class of trellis codes is defined based on a novel trellis structure. An algebraic basis is introduced using either the ring of integers Z M or the ring of polynomials P M defined over the primary field GF(p). Codes are constructed for M-PSK by mapping subgroups and their cosets onto branch planes in the trellis. Special attention is given to codes that are linear in the defined algebra and to those for which decoding is unaffected by certain phase errors in the receiver.

A mathematical framework for algorithm-based fault-tolerant computing over a ring of integers

Abstract: In this work, we establish a complete mathematical framework for algorithm-based fault-tolerant computing for data vectors defined over a ring of integers. The ring of integers consists of integers {0,1,…,M−1} and all the arithmetic operations are performed modulo the integerM, which is assumed to be composite. The importance of the work lies in the suitability of modulo arithmetic in certain computational environments. Lack of an underlying Galois field,GF(q), presents a unique challenge to this framework. We develop the theory and algorithms for single as well as multiple fault correction and detection. We also analyze the parallel and serial nature of the encoder and decoder configurations. Certain known but rather old results in the theory of numbers dealing with linear congruences and matrix algebra are also described and extended further using mathematical terminology that modern-day researchers are expected to be familiar with.

Residue Class Topologies

Abstract: A construction by the author of topologies on the rational field is generalized by considering the ring of rational integers as a special case of a discretely topologized subring of a field topologized by an absolute value.

On some applications of finitely generated semi-groups

Abstract: Let V be a finitely generated multiplicative semi-group with r generators in the ring of integers \(\mathbb{Z}_\mathbb{K}\)of an algebraic number field \(\mathbb{K}\)of degree n over ℚ. We use various bounds for character sums to obtain results on the distribution of the residues of elements of V modulo an integer ideal q. In the simplest case, when \(\mathbb{K} = \mathbb{Q}\)and r = 1 this is a classical question on the distribution of residues of an exponential function, which may be interpreted as concerning the quality of the linear congruential pseudo-random number generator. Besides this well known application we consider several other problems from algebraic number theory, the theory of function fields over a finite field, complexity theory, cryptography, and coding theory where results on the distribution of some group V modulo q play a central role.

Cyclic codes for T-user adder channel over integer rings

Abstract: Coding for the synchronous noiseless T-user real adder channel is considered by employing cyclic codes with symbols from an arbitrary finite integer ring. The code construction is based on the factorisation of x/sup n/-1 over the unit ring of an appropriate extension of a finite integer ring. Any number of users in the system can be independently active and the maximum achievable sum rate is 1 (when all T users are active).

Arithmetically independent integers and values of rational functions

Abstract: We prove the existence of arithmetically independent integers and their properties and apply them to values of rational functions on algebraic varieties.

On Hardy-Bessel potential spaces over the ring of integers in a local field

Abstract: Штрихарц [3] дал характе ристику пространствL p (R n ) бесселевых потенциа лов порядка α функций из п ространстваL p (R n ) с пом ощьюL p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern- ulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF α (P 0) (0 1/р−1) в терминах функц ионаловS α f(x) (1τ≤2), которые в случаеR n } соответствуютD α f (x), гдеР 0 - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов.

Method for improved security of crc encryption system

Abstract: The digital information is encrypted by first performing a preselected number of CRC iterations or partial convolutions by multiplication with a mask in the Galois Field. Before the CRC operation is completed, the intermediate resultant is subjected to an Integer Ring operation, such as addition, which injects a nonlinearity over the Galois Field due to possible arithmetic carry operations. After the Integer Ring operation, the Galois Field CRC process is continued to completion. The result is an encrypted value which is not readily decrypted by Galois Field techniques.

Bases normales d'entiers et unités modulaires

Abstract: LetK be an imaginary quadratic field with discriminantd K <−4,d K ≡2, 3 mod 4, andp a prime number,p≡1 mod 8,p split inK; let Ω p be the ring class field overK with conductorp andK(p) the ray class field overK with conductorp. An explicit normal basis is constructed for the ring of integers of the unique quadratic extension of Ω p contained inK(p) over the ring of integers of Ω p . This uses certain classical modular units considered by Deuring and Hecke.

Universal Positive Ternary Integral Quad. Forms over Real Quadratic Fields

Abstract: Maass [M] showed that the quadratic form x 2 +y 2 +z 2 is universal over the ring of integers of Q( p 5), i.e., it represents every totally positive integer in Q( p 5). In this paper, we extend this result to all real quadratic Þelds. We show that there are only three real quadratic Þelds which admit ternary universal classic integral quadratic forms; they are Q( p 2), Q( p 3) and Q( p 5). In each of these Þelds, we determine all ternary universal classic integral quadratic forms.