100,403 research outputs found
Efficient long division via Montgomery multiply
We present a novel right-to-left long division algorithm based on the
Montgomery modular multiply, consisting of separate highly efficient loops with
simply carry structure for computing first the remainder (x mod q) and then the
quotient floor(x/q). These loops are ideally suited for the case where x
occupies many more machine words than the divide modulus q, and are strictly
linear time in the "bitsize ratio" lg(x)/lg(q). For the paradigmatic
performance test of multiword dividend and single 64-bit-word divisor,
exploitation of the inherent data-parallelism of the algorithm effectively
mitigates the long latency of hardware integer MUL operations, as a result of
which we are able to achieve respective costs for remainder-only and full-DIV
(remainder and quotient) of 6 and 12.5 cycles per dividend word on the Intel
Core 2 implementation of the x86_64 architecture, in single-threaded execution
mode. We further describe a simple "bit-doubling modular inversion" scheme,
which allows the entire iterative computation of the mod-inverse required by
the Montgomery multiply at arbitrarily large precision to be performed with
cost less than that of a single Newtonian iteration performed at the full
precision of the final result. We also show how the Montgomery-multiply-based
powering can be efficiently used in Mersenne and Fermat-number trial
factorization via direct computation of a modular inverse power of 2, without
any need for explicit radix-mod scalings.Comment: 23 pages; 8 tables v2: Tweak formatting, pagecount -= 2. v3: Fix
incorrect powers of R in formulae [7] and [11] v4: Add Eldridge & Walter ref.
v5: Clarify relation between Algos A/A',D and Hensel-div; clarify
true-quotient mechanics; Add Haswell timings, refs to Agner Fog timings pdf
and GMP asm-timings ref-page. v6: Remove stray +bw in MULL line of Algo D
listing; add note re byte-LUT for qinv_
An Analysis of Arithmetic Constraints on Integer Intervals
Arithmetic constraints on integer intervals are supported in many constraint
programming systems. We study here a number of approaches to implement
constraint propagation for these constraints. To describe them we introduce
integer interval arithmetic. Each approach is explained using appropriate proof
rules that reduce the variable domains. We compare these approaches using a set
of benchmarks. For the most promising approach we provide results that
characterize the effect of constraint propagation. This is a full version of
our earlier paper, cs.PL/0403016.Comment: 44 pages, to appear in 'Constraints' journa
Algorithmic counting of nonequivalent compact Huffman codes
It is known that the following five counting problems lead to the same
integer sequence~: the number of nonequivalent compact Huffman codes of
length~ over an alphabet of letters, the number of `nonequivalent'
canonical rooted -ary trees (level-greedy trees) with ~leaves, the number
of `proper' words, the number of bounded degree sequences, and the number of
ways of writing with integers
. In this work, we show that one can
compute this sequence for \textbf{all} with essentially one power series
division. In total we need at most additions and
multiplications of integers of bits, , or bit
operations, respectively. This improves an earlier bound by Even and Lempel who
needed operations in the integer ring or bit operations,
respectively
Prime power terms in elliptic divisibility sequences
We consider a particular case of an analog for elliptic curves to the
Mersenne problem : finding explicitely all prime power terms in an elliptic
divisibility sequence when descent via isogeny is possible. We explain how this
question can be related to classical problems in diophantine geometry and we
compute an explicit upper bound on the index of prime power terms in magnified
elliptic divisibility sequences.Comment: 30 pages, submitte
Root finding with threshold circuits
We show that for any constant d, complex roots of degree d univariate
rational (or Gaussian rational) polynomials---given by a list of coefficients
in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a
uniform family of constant-depth polynomial-size threshold circuits). The basic
idea is to compute the inverse function of the polynomial by a power series. We
also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur
Number theoretic example of scale-free topology inducing self-organized criticality
In this work we present a general mechanism by which simple dynamics running
on networks become self-organized critical for scale free topologies. We
illustrate this mechanism with a simple arithmetic model of division between
integers, the division model. This is the simplest self-organized critical
model advanced so far, and in this sense it may help to elucidate the mechanism
of self-organization to criticality. Its simplicity allows analytical
tractability, characterizing several scaling relations. Furthermore, its
mathematical nature brings about interesting connections between statistical
physics and number theoretical concepts. We show how this model can be
understood as a self-organized stochastic process embedded on a network, where
the onset of criticality is induced by the topology.Comment: 4 pages, 3 figures. Physical Review Letters, in pres
Computing with and without arbitrary large numbers
In the study of random access machines (RAMs) it has been shown that the
availability of an extra input integer, having no special properties other than
being sufficiently large, is enough to reduce the computational complexity of
some problems. However, this has only been shown so far for specific problems.
We provide a characterization of the power of such extra inputs for general
problems. To do so, we first correct a classical result by Simon and Szegedy
(1992) as well as one by Simon (1981). In the former we show mistakes in the
proof and correct these by an entirely new construction, with no great change
to the results. In the latter, the original proof direction stands with only
minor modifications, but the new results are far stronger than those of Simon
(1981). In both cases, the new constructions provide the theoretical tools
required to characterize the power of arbitrary large numbers.Comment: 12 pages (main text) + 30 pages (appendices), 1 figure. Extended
abstract. The full paper was presented at TAMC 2013. (Reference given is for
the paper version, as it appears in the proceedings.
Generalized Ehrhart polynomials
Let be a polytope with rational vertices. A classical theorem of Ehrhart
states that the number of lattice points in the dilations is a
quasi-polynomial in . We generalize this theorem by allowing the vertices of
P(n) to be arbitrary rational functions in . In this case we prove that the
number of lattice points in P(n) is a quasi-polynomial for sufficiently
large. Our work was motivated by a conjecture of Ehrhart on the number of
solutions to parametrized linear Diophantine equations whose coefficients are
polynomials in , and we explain how these two problems are related.Comment: 18 pages, no figures; v2: Sections 4 and 5 added, proofs and
exposition have been expanded and clarifie
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