Number Theoretic Algorithms Research Articles

Degree and height estimates for modular equations on PEL Shimura varieties

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and implies complexity bounds for number-theoretic algorithms using these modular equations. In particular, we obtain tight degree bounds for modular equations of Siegel and Hilbert type for abelian surfaces.

Study of the k-ary GCD Algorithm

In our paper, we consider some approaches to accelerating the calculation of the Greatest Common Divisor (GCD) of natural numbers for given A and B based on the k-ary GCD algorithm. Such a task arises in the implementation of many mathematical and number-theoretic algorithms, for example, in cryptography and number theory (see [8]).

Об оценках сложности алгоритмов извлечения квадратных корней в конечных полях и кольцах вычетов

The article presents a review of the known algorithms for extracting square roots of the quadratic residues modulo primes and prime powers. According to the Chinese remainder theorem, such algorithms can be constructed to calculate square roots modulo any integer. These include the Tonelli and Chipolla probabilistic algorithms and deterministic algorithms for calculating the square roots of quadratic residues in residue rings modulo a prime number not congruent to 1 modulo 8, as well as the algorithms for extracting a square root modulo a power of a prime number constructed on their basis. The article presents new algorithms for extracting square roots in finite fields of characteristic congruent to 3 modulo 4, for square roots of residues congruent to 1 modulo 8, and for square roots in residue rings modulo power of two. For probabilistic algorithms, the calculation complexity and the probability of successfully accomplishing the calculation in the first try are estimated. The square roots extraction problem encountered in number-theoretic algorithms of cryptography used in combination with the Chinese remainder theorem for root square extraction in residue rings modulo an arbitrary number is considered. The article presents known probabilistic and deterministic algorithms for extracting square roots modulo a prime number, along with estimating the complexity of the algorithms and the probability of successfully completing the computation in the first try. New implementing in the elliptic curve cryptography algorithms for extracting square roots in finite fields of characteristic congruent to 3 modulo 4 are substantiated, and estimates of their complexity are given. Algorithms for extracting square roots in fields of even order are studied, and estimates of their complexity are given. Algorithms for extracting square roots in residue rings modulo prime powers and modulo power of two are demonstrated.

Parallel implementations of Brunotte’s algorithm

In this paper the author make a comprehensive comparison of different parallelizations of a sequential number theoretic algorithm having large memory requirements. Brunotte’s algorithm is one of the currently known best methods for the decision of the canonical number system (or more generally shift radix system) property. Still, it can be very space-consuming in some cases. Pushing the algorithm to its limits may hopefully shed light on mathematical patterns that would otherwise not be discernible. The algorithm contains many n -dimensional vector operations and set operations like insert, find, clear, etc. The parallel algorithms encounter two difference kinds of concurrency problems. First, they need computationally intensive arithmetic vector operations, second, the set implementations require a huge amount of memory and general purpose processors. The algorithms described in this article are basically designed for two platforms. The first platform is a generic symmetric multiprocessing (SMP) architecture without any vector processor extension, the second is the Cell Broadband Engine. The SMP platforms have several general purpose processors in contrast with the Cell Broadband Engine where the processors have Synergistic vector processors.

Diophantine frequency synthesis

A methodology for fine-step, fast-hopping, low-spurs phase-locked loop based frequency synthesis is presented. It uses mathematical properties of integer numbers and linear Diophantine equations to overcome the constraining relation between frequency step and phase-comparator frequency that is inherent in conventional phase-locked loop based frequency synthesis. The methodology leads to fine-step, fast-hopping, modular-structured frequency synthesizers with potentially very low spurs, especially in the vicinity of the carrier. The paper focuses on the mathematical principles of the new methodology and the related number theoretic algorithms.

Detecting perfect powers by factoring into coprimes

This paper presents an algorithm that, given an integer n > 1, finds the largest integer k such that n is a kth power. A previous algorithm by the first author took time b 1+o(1) where b = Ign; more precisely, time b exp(O(lg b lg lg b)); conjecturally, time b(lg b) O(1) . The new algorithm takes time b(lg b) O(1) . It relies on relatively complicated subroutines-specifically, on the first author's fast algorithm to factor integers into coprimes-but it allows a proof of the b(lg b) O(1) bound without much background; the previous proof of b 1+o(1) relied on transcendental number theory. The computation of k is the first step, and occasionally the bottleneck, in many number-theoretic algorithms: the Agrawal-Kayal-Saxena primality test, for example, and the number-field sieve for integer factorization.

Threshold-based declustering

Declustering techniques reduce query response time through parallel I/O by distributing data among multiple devices. Except for a few cases it is not possible to find declustering schemes that are optimal for all spatial range queries. As a result of this, most of the research on declustering has focused on finding schemes with low worst case additive error. However, additive error based schemes have many limitations including lack of progressive guarantees and existence of small non-optimal queries. In this paper, we take a different approach and propose threshold-based declustering. We investigate the threshold k such that all spatial range queries with ⩽ k buckets are optimal. Upper bound on threshold is analyzed using bound diagrams and a number theoretic algorithm is proposed to find schemes with high threshold value. Threshold-based schemes have many advantages: they have low worst-case additive error, provide progressive guarantees by dividing larger queries into subqueries with ⩽ k buckets, can be used to compare replicated declustering schemes and render many large complementary queries optimal.

Number-theoretic approach to optimum velocity decoding given quantized position information

The problem of optimum velocity estimation, given a sequence of outputs from a digital position sensor, is important in many process applications and motion control systems. The sequence obtained when the output of a constant rate system is quantized, and then uniformly sampled, can be represented as a nonhomogeneous spectrum. An analogous sequence finds application in computer graphics in problems such as the digitization of straight lines. In this paper, the number-theoretic framework developed to represent graphic systems is modified for application to velocity sensing. The assumption of close-to-constant velocity often holds true in high-inertia and regulator-type applications. To cater for periods when this assumption is invalid, the velocity estimation algorithms described include an optimum, linear finite-impulse response (FIR) differentiator. Both on-line and off-line (look-up table based) versions of the algorithm are presented. The mean-squared error (MSE) associated with the number-theoretic algorithm is shown to be superior to that of linear filters. This is experimentally demonstrated through using digital signal processor (DSP)-based instrumentation applied to a motion-control-based test-rig. The new filter is also applicable to other areas in which quantized signals are differentiated.

A fast and robust RNS algorithm for evaluating signs of determinants

A new and efficient number theoretic algorithm for evaluating signs of determinants is proposed. The algorithm uses computations over small finite rings. It is devoted to a variety of computational geometry problems, where the necessity of evaluating signs of determinants of small matrices often arises.

Batch RSA

Number theoretic cryptographic algorithms are all based upon modular multiplication modulo some composite or prime. Some security parameter n is set (the length of the composite or prime). Cryptographic functions such as digital signature or key exchange require O(n) or O(?n) modular multiplications ([DH, RSA, R, E, GMR, FS], etc.).This paper proposes a variant of the RSA scheme which requires only polylog(n) (O(log2 n)) modular multiplications per RSA operation. Inherent to the scheme is the idea of batching, i.e., performing several encryption or signature operations simultaneously. In practice, the new variant effectively performs several modular exponentiations at the cost of a single modular exponentiation. This leads to a very fast RSA-like scheme whenever RSA is to be performed at some central site or when pure-RSA encryption (vs. hybrid encryption) is to be performed.An important feature of the new scheme is a practical scheme that isolates the private key from the system, irrespective of the size of the system, the number of sites, or the number of private operations that need be performed.

Rings, fields, the Chinese remainder theorem and an extension-Part I: theory

The much celebrated Chinese Remainder Theorem has been widely employed in designing fast computationally efficient algorithms in the field of digital signal processing. It has two versions. One is over a ring of integers and the second is over a ring of polynomials with Coefficients defined over a field. In this research work, we extend the Chinese Remainder Theorem to the case of a ring of polynomials with coefficients defined over a finite ring of integers. The entire work is closely related to the already established results on finite fields. This extension is expected to serve as a keystone in the future design of number-theoretic algorithms for performing some of the most computationally intensive tasks. This approach is superior to the number-theoretic-transforms in the sense that the limitations on both the word length and the sequence length are completely removed. In fact, the number-theoretic-transforms may be considered as a very special case of our general approach. Furthermore, the computations required in this work. Which inherits all the merits of the Chinese Remainder Theorem, can be performed in parallel. >

Explicit Bounds for Primality Testing and Related Problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is $O({\log ^2}m)$. This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most $2{\log ^2}m$.

Number theoretic fast algorithms for bilinear and other generalized transformations

Fast algorithms based on the Mersenne and Fermat number-theoretic transforms are used to perform the bilinear transformation of a continuous transfer function to a discrete equivalent. The computations are carried out in finite precision arithmetic, require no multiplications, and can be implemented in parallel using very simple processors. Although the bilinear transform is presently emphasized, similar algorithms are easily derived for any transformation from the s-plane to the z-plane involving the ratio of two polynomials with integer coefficients.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Explicit bounds for primality testing and related problems

Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits a number that is O ( log 2 m ) O({\log ^2}m) . This has been generalized by Lagarias, Montgomery, and Odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields. This paper gives a different proof of this theorem, in which explicit constants are supplied. The bounds imply that if the ERH holds, a composite number m has a witness for its compositeness (in the sense of Miller or Solovay-Strassen) that is at most 2 log 2 m 2{\log ^2}m .

Analytic Methods in the Analysis and Design of Number-Theoretic Algorithms.

This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoretic algorithms and to the construction of cryptographic protocols.Contents: Introduction; 1: Explicit Bounds for Primality Testing; Ankeny's Theorem and its Algorithmic Consequences; Background from Number Theory; Roots; Asymptotic Theorems; Zeta-function Estimates; Numerical Theorems; Computing Bounds for Specific Moduli; Comparisons with Empirical Results; 2: The Generation of Random Factorizations; Introduction; A Method That Almost Works; Doctoring the Odds; A Factor Generation Procedure; The Complete Algorithm; 2.5 Bounds for the Number of Prime Tests; A Single-precision Time Bound; The Use of Probabilistic Primality Tests.Eric Bach received his doctorate from the University of California at Berkeley. He is currently an Assistant Professor of Computer Science at the University of Wisconsin at Madison. Analytic Methods in the Analysis and Design of Number Theoretic Algorithms is a 1984 ACM Distinguished Dissertation.