Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

We examine the problem of factoring the th cyclotomic polynomial, over , and distinct primes. Given the traces of the roots of we construct the coefficients of in time . We demonstrate a deterministic algorithm for factoring in time when has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of in time .

     

Sharpening ``Primes is in P' for a large family of numbers

We present algorithms that are deterministic primality tests for a large family of integers, namely, integers for which an integer is given such that the Jacobi symbol , and integers for which an integer is given such that . The algorithms we present run in time, where is the exact power of dividing when and if . The complexity of our algorithms improves up to when . We also give tests for a more general family of numbers and study their complexity.

     

On iterates of Möbius transformations on fields

Let be a quadratic polynomial over a splitting field , and be the set of zeros of . We define an associative and commutative binary relation on so that every Möbius transformation with fixed point set is of the form ``plus' for some . This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of are distinct and the iterates of converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.

     

Computation of relative class numbers of CM-fields by using Hecke -functions

We develop an efficient technique for computing values at of Hecke -functions. We apply this technique to the computation of relative class numbers of non-abelian CM-fields which are abelian extensions of some totally real subfield . We note that the smaller the degree of the more efficient our technique is. In particular, our technique is very efficient whenever instead of simply choosing (the maximal totally real subfield of ) we can choose real quadratic. We finally give examples of computations of relative class numbers of several dihedral CM-fields of large degrees and of several quaternion octic CM-fields with large discriminants.

     

Computer verification of the Ankeny-Artin-Chowla Conjecture for all primes less than

Let be a prime congruent to 1 modulo 4, and let be rational integers such that is the fundamental unit of the real quadratic field . The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that will not divide . This is equivalent to the assertion that will not divide , where denotes the th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers ; for example, when , then both and exceed . In 1988 the AAC conjecture was verified by computer for all . In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes up to .

     

On the modular curves

Let denote an elliptic curve over and the modular curve classifying the elliptic curves over such that the representations of in the 7-torsion points of and of are symplectically isomorphic. In case is given by a Weierstraß equation such that the invariant is a square, we exhibit here nontrivial points of . From this we deduce an infinite family of curves for which has at least four nontrivial points.

     

Error estimates for the three-field formulation with bubble stabilization

In this paper we prove convergence and error estimates for the so-called 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in and in the broken norm for the internal variable , and in suitable weighted norms for the other variables and .     

Wavelet bases in

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .

     

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of

     

Period of the power generator and small values of Carmichael's function

Consider the pseudorandom number generator where we are given the modulus , the initial value and the exponent . One case of particular interest is when the modulus is of the form , where are different primes of the same magnitude. It is known from work of the first and third authors that for moduli , if the period of the sequence exceeds , then the sequence is uniformly distributed. We show rigorously that for almost all choices of it is the case that for almost all choices of , the period of the power generator exceeds . And so, in this case, the power generator is uniformly distributed.

We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto.

The principal tool is an estimate related to the Carmichael function , the size of the largest cyclic subgroup of the multiplicative group of residues modulo . In particular, we show that for any , we have for all integers with , apart from at most exceptions.

     

Korovkin tests, approximation, and ergodic theory

We consider sequences of matrices with a block structure spectrally distributed as an -variate matrix-valued function , and, for any , we suppose that is a linear and positive operator. For every fixed we approximate the matrix in a suitable linear space of matrices by minimizing the Frobenius norm of when ranges over . The minimizer is denoted by . We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:

1.

the sequence is distributed as ,

2.

the sequence is distributed as the constant function (i.e. is spectrally clustered at zero).

The first result is an ergodic one which can be used for solving numerical approximation theory problems. The second has a natural interpretation in the theory of the preconditioning associated to cg-like algorithms.

     

Upper bounds for the prime divisors of Wendt's determinant

Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.

     

Two efficient algorithms for the computation of ideal sums in quadratic orders

This paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders. If the class number of the quadratic number field is equal to 1, these algorithms can be used to calculate the GCD in the quadratic order. We show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in up to a constant factor, i.e., in where bounds the size of the operands and denotes an upper bound for the multiplication time of -bit integers. Using Schönhage-Strassen's asymptotically fast multiplication for -bit integers, we achieve

     

Efficient CM-constructions of elliptic curves over finite fields

We present an algorithm that, on input of an integer together with its prime factorization, constructs a finite field and an elliptic curve over for which has order . Although it is unproved that this can be done for all , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in , where is the number of distinct prime factors of . In the cryptographically relevant case where is prime, an expected run time can be achieved. We illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order and nextprime.

     

Higher-order Carmichael numbers

We define a Carmichael number of order to be a composite integer such that th-power raising defines an endomorphism of every -algebra that can be generated as a -module by elements. We give a simple criterion to determine whether a number is a Carmichael number of order , and we give a heuristic argument (based on an argument of Erdos for the usual Carmichael numbers) that indicates that for every there should be infinitely many Carmichael numbers of order . The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order .

     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Finding strong pseudoprimes to several bases

Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pomerance et al. and Jaeschke, are known for . Upper bounds for were given by Jaeschke.

In this paper we tabulate all strong pseudoprimes (spsp's) to the first ten prime bases which have the form with odd primes and There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp's to both bases 31 and 37. As a result the upper bounds for and are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke's and Arnault's methods are given.

     

The Postage Stamp Problem: An algorithm to determine the

Given an integral ``stamp" basis with and a positive integer , we define the -range as

. For given and , the extremal basis has the largest possible extremal -range

We give an algorithm to determine the -range. We prove some properties of the -range formula, and we conjecture its form for the extremal -range. We consider parameter bases , where the basis elements are given functions of . For we conjecture the extremal parameter bases for .

     

On searching for solutions of the Diophantine equation

We propose an efficient search algorithm to solve the equation for a fixed value of 0$">. By parametrizing , this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of below 1000 for which no solution had previously been found. We found three new integer solutions for and 931 in the range of .

     

Fast algorithms for discrete polynomial transforms

Consider the Vandermonde-like matrix , where the polynomials satisfy a three-term recurrence relation. If are the Chebyshev polynomials , then coincides with . This paper presents a new fast algorithm for the computation of the matrix-vector product in arithmetical operations. The algorithm divides into a fast transform which replaces with and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes with almost the same precision as the Clenshaw algorithm, but is much faster for .