Showing papers on "Integer published in 1985"

Efficient Implementations of the Chinese Remainder Theorem for Sign Detection and Residue Decoding

Abstract: Two conversion techniques based on the Chinese remainder theorem are developed for use in residue number systems. The new implementations are fast and simple mainly because adders modulo a large and arbitrary integer M are effectively replaced by binary adders and possibly a lookup table of small address space. Although different in form, both techniques share the same principle that an appropriate representation of the summands must be employed in order to evaluate a sum modulo M efficiently. The first technique reduces the sum modulo M in the conversion formula to a sum modulo 2 through the use of fractional representation, which also exposes the sign bit of numbers. Thus, this technique is particularly useful for sign detection and for any operation requiring a comparison with a binary fraction of M. The other technique is preferable for the full conversion from residues to unsigned or 2's complement integers. By expressing the summands in terms of quotients and remainders with respect to a properly chosen divisor, the second technique systematically replaces the sum modulo M by two binary sums, one accumulating the quotients modulo a power of 2 and the other accumulating the remainders the ordinary way. A final recombination step is required but is easily implemented with a small lookup table and binary adders.

Correlation Immunity and the Summation Generator

Abstract: It is known that for a memoryless mapping from GF(2)N into GF(2) the nonlinear order of the mapping and its correlation-immunity form a linear tradeoff. In this paper it is shown that the same tradeoff does no longer hold when the function is allowed to have memory. Moreover, it is shown that integer addition, when viewed over GF(2), defines an inherently nonlinear function with memory whose correlation-immunity is maximum. The summation generator which sums N binary sequences over the integers is shown as an application of integer addition in random sequence generation.

Randomized Rounding: A Technique for Provably Good Algorithms and

Abstract: We study the relation between a class of 0-1 integer linear programs and their rational relaxations. We show that the rational optimum to a problem instance can be used to construct a provably good 0-1 solution by means of a randomized algorithm. Our technique can be extended to provide bounds on the disparity between the rational and 0-1 optima for a given problem instance.

Shift-Register Synthesis (Modulo m)

Abstract: The Berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. We present an algorithm which generalizes Berlekamp’s to the case when the elements of the sequence are integers modulo m, where m is an arbitrary (but known) integer. Details will be published elsewhere.

On Hilbert's 17th problem and real Nullstellensatz for global analytic functions

Abstract: The author proves a Nullstellensatz for the ring of real analytic functions on a compact analytic manifold. The main results are the following. Theorem 1: Let X be a compact irreducible analytic set of a real analytic manifold M and f:X→R a nonnegative analytic function. Then f is a sum of squares of meromorphic functions. Theorem 2: Let I be a finitely generated ideal of O(M) with Z(I) compact. Then IZ(I)=I√R, where Z(I) is the zero set of I, IZ(I) the ideal of functions (in O(M)) vanishing on I, and I√R the real radical of I (i.e. the set of functions f in O(M) such that there exist g1,⋯,gk and an integer p with f2p + g2 1 + ⋯ +g2k ∈ I). Corollary: Let I be as in Theorem 2. Then IZ(I)=I if and only if I is real (i.e. I=I √ R). The proofs are based on results about extension of orders.

Comments on "A Computer Algorithm for Calculating the Product AB Modulo M"

Abstract: The modular multiplication algorithm (MMA) was presented as a method of calculating " the smallest nonnegative integer R congruent modulo M to the product AB of two nonegative integers without dividing by M."1 The claim that division is avoided is technically correct, but misleading. A minor modification calculates both R and Q such that AB = MQ + R. A simplified version of the new algorithm is given and an alternate derivation is shown to illustrate the key ideas behind the method.

Integer points on curves and surfaces

Abstract: Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ℝ3 which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ℝn with nonvanishing Gaussian curvature.

Encryption apparatus and methods for raising a large unsigned integer to a large unsigned integer power modulo a large unsigned integer

Abstract: There is disclosed an encryption apparatus which apparatus functions to raise a large unsigned integer (B) indicative of message data to a large unsigned integer power E, modulo a large unsigned integer M with each of said integers being as large as N bits wherein the resulting large unsigned integer C is adapted for transmission over an insecure communications channel. The apparatus may likewise operate on a received integer C to recover the decrypted message B. The circuitry includes first logic means which is responsive to the large unsigned integer (B) for successively squaring said integer including means for reducing said squared integers successively by a given modulus M to provide at an output a first value indicative of said squared integer as reduced by said given modulus, selectively operated gating means are coupled to said first logic means and operates to receive bits of a given exponent power E. The gating means applies selected bits of the exponent power to second logic means which are controlled by said gating means to provide the product of said first value as further modified by modulus means for providing at its output the large unsigned integer for transmission over the insecure communications channel. The first and second logic means are constructed such that the required circuit size to perform the computation is significantly reduced in complexity resulting in increased speed together with a substantial reduction in cost.

Groups with a small covering number

Abstract: The covering number of a group G is the smallest integer k, such that Ck=G for all nonidentity conjugacy classes C of G. If no such integer exists, the covering number is defined to be ω, the smallest infinite ordinal. In this article the frequency of groups with covering number equal to two is studied. While such finite groups are rare, there are many natural examples of such infinite groups.

A systolic algorithm for integer GCD computation

Abstract: It is shown that the greatest common divisor of two n-bit integers (given in the usual binary representation) can be computed in time O(n) on a linear systolic array of O(n) identical cells.

Proof of hayman's conjecture on normal families

Abstract: In 1964, Hayman posed the following conjecture. Let a(≠0) and b be two finite complex numbers and suppose n(≥5) be a positive integer. If is a family of meromorphic functions in a domain D and for each f∈ and z∈D, there exists f'(z)—af(z) n ≠b, then is normal in D. This paper aims at giving a proof of the conjecture.

Normal ordering and generalised Stirling numbers

Abstract: The relation between some normal ordering forms for boson operator functions and generalised Stirling numbers is shown. In particular, the ordering of the operator function ak(ar+N+s)n is obtained for positive integers k, r, n and an arbitrary integer s. This is a generalisation of the recent result of Katriel (ibid., vol.16, p.4171-3, 1983). Some other normal ordering formulae are presented. It was shown how to obtain antinormal forms from several normal expansions given here.

Localizations of injective modules

Abstract: The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

The parallel complexity of exponentiating polynomials over finite fields

Abstract: Modular integer exponentiation (given a, e, and m, compute ae mod m) is a fundamental problem in algebraic complexity for which no efficient parallel algorithm is known. Two closely related problems are modular polynomial exponentiation (given a(x), e, and m(x), compute (a(x))e mod m(x)) and polynomial exponentiation (given a(x), e. and t, compute the coefficient of xt in (a(x))e). It is shown that these latter two problems are in NC2 when a(x) and m(x) are polynomials over a finite field whose characteristic is polynomial in the input size.

On Decomposing Graphs into Isomorphic Uniform 2-Factors

Abstract: Publisher Summary For v, an even integer, let Hv be the complete graph on v vertices with the edges of a 1 - factor deleted and, for v odd, let Hv be the complete graph on v vertices. The Oberwolfach problem is to determine whether, for any given 2-factor G of Hv, where v is odd, it is possible to decompose Hv into 2 - factors, each of which is isomorphic to G. The corresponding problem when v is even is called the spouse-avoiding Oberwolfach problem. Thus, the chapter investigates special problems of these kinds.

Computing an arithmetic constant related to the ring of Gaussian integers

Abstract: We compute the analogue for Z[ i] of Euler's constant, that is 8 = limn where an = (?22.k6.n 1/frk2) log n. For this purpose we give an estimate for rk = min{ r > 0; there exists z E C such that card(Z[i ] n iD (z, r)) > k and we compute a great number of values of rk. 0. Results and Notations. Let Z[i] = (a + ib e C; a, b E Z} be the ring of Gaussian integers. For each integer k > 2 set rk = mint r > 0; there exists z e C such that card(Z[i] n Di(z, r)) > k }, where D(z, r) = { w e C; I w zI < r } is the closed disk with center z and radius r. Clearly, this minimum is attained and a closed disk with radius rk containing at least k integer (i.e., Gaussian integer) points will be called a minimal disk Dk and its boundary rk will be called a minimal circle. It is not known whether all the minimal disks with index k are deduced from one another by an isometry of R2 _ C which stabilizes the lattice Z[i]. Nevertheless, Sections 1 and 2 deal with some properties of these disks. In particular, we prove in Section 2 an estimate of rk from which we deduce the existence of 8 = limo n ,n where Sn= -logn. 2

Words periodic over the center of a division ring

Abstract: In generalization of a result of Herstein, the authors prove that, in a division ring with uncountable center, if any given nontrivial group word takes only values periodic over the center, then the division ring is commutative. Techniques include use of the result that a noncommutative division ring finite-dimensional over its center includes a nonabelian free group in its multiplicative group. The first author has shown [7] that a noncommutative division ring with uncountable center contains a nonabelian free subsemigroup in its multiplicative group. In attempting to generalize his methods to prove the existence of a free subgroup, we instead derive a result limiting the existence of a group word, all of whose values over a division ring are periodic over the center. Specifically, we prove that no nontrivial example of such a word can exist for a noncommutative division ring with uncountable center. This generalizes a result of Herstein [2], which considers the specific case of the multiplicative commutator (as a group word). Our methods of proof include the use of a polynomial identity to reduce to the case of a finite-dimensional divisional ring and the use of a form of the Tits alternative in such a case. The form we will use states that in a division ring finite-dimensional over its center, the multiplicative group either contains a nonabelian group or is commutative (i.e. we have a field). This is implicit in the work of Lichtman [5 and 6] (or see [1]). Or we may outline an argument as follows: consider the multiplicative group as a linear group over the center. Then by the original Tits alternative for fields [9], either the group contains a free subgroup or is solvable-by-locally finite (see Tits' Theorem 2(ii) for nonzero characteristic, or use Theorem 1 and substitute the intersection of the conjugates for zero characteristic). Now the normal solvable subgroup is central by Scott's result [8]. Then the division ring is locally finite over the center, and a result of Kaplansky [4] implies that the division ring is a field. We now discuss our definitions. By the term group word we simply mean an element w of a free group, say on some set of generators. The values of a group word over a division ring are the results of substituting nonzero elements of the division ring for each of the generators appearing in the word. An element d of a division ring is said to be periodic over a subset S of the division ring if d ' Ee S for some integer n > 0. Received by the editors April 27, 1984 and, in revised form, August 10, 1984. 1980 Mathematics Subject Classification. Primary 16A39; Secondary 16A70. ' This paper was presented at AMS Meeting # 810, Special Session in Ring Theory, April 6, 1984. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Discrete cells properties in the boundary set setting

Abstract: Let X be the complement of a a-Z-set in a locally compact separable ANR. It is proved that X satisfies the discrete n-cells property for each nonnegative integer n if and only if X satisfies the discrete approximation property. As a consequence, Hubert space manifolds that arise as complements of boundary sets in Hubert cube manifolds are characterized in terms of their homological structure coupled with a minimal amount of general positioning. 1. Introduction. H. Torunczyk characterizes those infinite-dimensional manifolds modeled on the Hubert cube and Hilbert space entirely in terms of geometric general position properties. Hilbert cube manifolds are precisely those locally compact ANR's X for which arbitrary pairs of maps of the Hilbert cube into A" are approximable by maps whose images are disjoint (Toi). This general position property easily is seen to be equivalent to the disjoint «-cells property (maps of the «-cell into X are approximable by maps with disjoint images) holding simultaneously for all nonnegative integers n. Hilbert space manifolds are precisely those topologi- cally complete separable ANR's X for which arbitrary sequences of maps of the Hilbert cube into X are strongly approximable by maps whose images form a discrete family (To2, BBMW). This latter general position property is known as the discrete approximation property, and the question naturally arises as to whether or not this property is equivalent to the analogous «-dimensional properties holding simultaneously for all nonnegative integers «. Recently, M. Bestvina, J. Mogilski, J. Walsh, and the author constructed examples of topologically complete separable ANR's that satisfy the discrete «-cells property for each nonnegative integer «, but fail to satisfy the discrete approximation property (BBMW). (Precise definitions of the aforementioned terms are found in §2.) None of the examples constructed in (BBMW) arise as complements of a-Z-sets in locally compact separable ANR's, a common source of Hilbert space manifolds. Our Main Result appears in §3 and shows that no such examples exist in this setting, henceforth referred to as the "boundary set setting". See (Cu). Main Result. Let F be a a-Z-set in the locally compact separable ANR Y. Then X = Y — F satisfies the discrete n-cells property for each nonnegative integer n if and only if X satisfies the discrete approximation property.

Pseudorandom number generation and space complexity

Abstract: Recently, Blum and Micali [3] described a pseudorandom number generator that transforms each m-bit seed to an ink-bit pseudorandom number, for any integer k. Under the assumption that the discrete logarithm problem cannot be solved by any polynomial-size combinational logic circuit, they show that the pseudorandom numbers generated are good in the sense that no polynomial-size circuit can determine the t th bit given the I st through ( t l ) st bits, with better than 50% accuracy. Yao [12] has shown, under the same assumption about the nonpolynomial complexity of the discrete logarithm problem, that these psuedorandom numbers can be used in place of truly random numbers by any polynomial-time probabilistic Turing machine. Thus, given a time n k probabilistic Turing machine M and given any e > 0, a deterministic Turing machine can simulate M by cycling through all seeds of length n*, giving a deterministic simulation in time 2 he, an improvement over the time 2 nk taken by the obvious simulation. Yao also shows that other problems, for example, integer factorization, can be used instead of the discrete logarithm in the intractability assumption.

Integer-valued self-similar processes

Abstract: Let IN0 denote the set of nonnegative integers. We consider IN0-valued analogues of self-similar processes by defining Unvalued fractions of IN0-valued processes. These fractions are defined in terms of sums of independent Markov branching processes, in such a way that the one-dimensional marginals coincide with the IN0-valued multiples of IN0-valued random variables as introduced in [10] and [3]; this requirement still leaves room for several definitions of an 0-valued fraction, and a sensible choice has to be made. The relation with branching processes has two aspects. On the one hand, results from the theory of these processes can be used to prove analogues of classical theorems, on the other hand new results about branching processes are suggested by translating analogues of classical results in terms of branching processes. In this way, analogues are derived of the basic properties of classically self-similar processes such as obtained by Lamperti [5], and a simple relationship is established between...

Averaging effects on irregularities in the distribution of primes in arithmetic progressions

Abstract: Let t be an integer taking on values between 1 and x (x real), let ni,bc(t) denote the number of positive primes 2 to quadratic residues of b be y(b) to 1, and let

A general character to integer conversion method

Abstract: Most programs which involve character to integer conversion fail to detect integer overflow correctly. This paper describes some of the pitfalls of not recognizing overflow, and presents a correct method for doing so in an arbitrary base for any precision.

On an asymptotic formula for the niven numbers

Abstract: A Niven number is a positive integer which is divisible by its digital sum. A discussion of the possibility of an asymptotic formula for N(x) is given. Here, N(x) denotes the nmber of Niven numbers less than x. A partial result will be presented. This result will be an asymptotic formula for Nk(x) which denotes the number of Niven numbers less than x with digital sum k.