Three- and four-dimensional -optimal lattice rules of moderate trigonometric degree

A systematic search for optimal lattice rules of specified trigonometric degree over the hypercube has been undertaken. The search is restricted to a population of lattice rules . This includes those where the dual lattice may be generated by points for each of which . The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for , and , , a list of -optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.

     

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime we show a way to construct large families of polynomials , , where is a set of primes of the form mod and is the irreducible polynomial of the Gaussian periods of degree in . Examples of these families when are worked in detail. We also show, given an integer and a prime mod , how to represent by matrices the Gaussian periods of degree in , and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of .

     

Primitive central idempotents of finite group rings of symmetric groups

Let be a prime. We denote by the symmetric group of degree , by the alternating group of degree and by the field with elements. An important concept of modular representation theory of a finite group is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring , where is a prime power. Here, we describe a new method to compute the primitive central idempotents of for arbitrary prime powers and arbitrary finite groups . For the group rings of the symmetric group, we show how to derive the primitive central idempotents of from the idempotents of . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of which contains the primitive central idempotents. The described results are most efficient for . In an appendix we display all primitive central idempotents of and for which we computed by this method.

     

A hidden number problem in small subgroups

Boneh and Venkatesan have proposed a polynomial time algorithm for recovering a hidden element , where is prime, from rather short strings of the most significant bits of the residue of modulo for several randomly chosen . González Vasco and the first author have recently extended this result to subgroups of of order at least for all and to subgroups of order at least for almost all . Here we introduce a new modification in the scheme which amplifies the uniformity of distribution of the multipliers and thus extend this result to subgroups of order at least for all primes . As in the above works, we give applications of our result to the bit security of the Diffie-Hellman secret key starting with subgroups of very small size, thus including all cryptographically interesting subgroups.

     

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.

     

Prime decomposition in the anti-cyclotomic extension

For an imaginary quadratic number field and an odd prime number , the anti-cyclotomic -extension of is defined. For primes of , decomposition laws for in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of is -embeddable. For some and , we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo . The article contains many numerical examples.

     

The Lucas-Pratt primality tree

In 1876, E. Lucas showed that a quick proof of primality for a prime could be attained through the prime factorization of and a primitive root for . V. Pratt's proof that PRIMES is in NP, done via Lucas's theorem, showed that a certificate of primality for a prime could be obtained in modular multiplications with integers at most . We show that for all constants , the number of modular multiplications necessary to obtain this certificate is greater than for a set of primes with relative asymptotic density 1.

     

Optimal logarithmic energy points on the unit sphere

We study minimum energy point charges on the unit sphere in , , that interact according to the logarithmic potential , where is the Euclidean distance between points. Such optimal -point configurations are uniformly distributed as . We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order . Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term in the asymptotical expansion of the optimal energy. Previously, this was known for the unit sphere in only. Furthermore, we present an upper bound for the error of integration for an equally-weighted numerical integration rule with the nodes forming an optimal logarithmic energy configuration. For polynomials of degree at most this bound is as . For continuous functions of satisfying a Lipschitz condition with constant the bound is as .

     

Computation of several cyclotomic Swan subgroups

Let denote the locally free class group, that is the group of stable isomorphism classes of locally free -modules, where is the ring of algebraic integers in the number field and is a finite group. We show how to compute the Swan subgroup, , of when , a primitive -th root of unity, , where is an odd (rational) prime so that and 2 is inert in We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ring; we do the computations obtaining for several primes a nontrivial divisor of These calculations give an alternative proof that the fields for =11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

     

Sums of heights of algebraic numbers

For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

     

Polyharmonic splines on grids and their limits

Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines on such grids for the limiting process 0.$"> For a large class of data functions defined on it turns out that there exists a limit function This limit function is shown to be a polyspline of order on strips. By the theory of polysplines we know that the function is smooth up to order everywhere (in particular, they are smooth on the hyperplanes , which includes existence of the normal derivatives up to order while the RBF interpolants are smooth only up to the order The last fact has important consequences for the data smoothing practice.

     

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree can interpolate given points in is confirmed for under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order can be achieved as soon as the interpolating curve exists.

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

On the uniformity of distribution of the RSA pairs

Let be a product of two distinct primes and . We show that for almost all exponents with the RSA pairs are uniformly distributed modulo when runs through

the group of units modulo (that is, as in the classical RSA scheme);

the set of -products , , where are selected at random (that is, as in the recently introduced RSA scheme with precomputation).

These results are based on some new bounds of exponential sums.

     

A solution to certain polynomial equations with applications to nonlinear fitting

We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in variables by reducing it to the problem of solving two special linear systems of size and rooting a single univariate polynomial of degree . Over , all solutions can be found with fixed precision using, up to polylogarithmic factors, bitwise operations in the worst case. Furthermore, if the data is well conditioned, then this can be reduced to bit operations, up to polylogarithmic factors. As an application, we show how this can be used to fit data to a complex exponential sum with terms in the same, nearly optimal, time.

     

Constructing representations of higher degrees of finite simple groups and covers

Let be a finite group and an irreducible character of . A simple method for constructing a representation affording can be used whenever has a subgroup such that has a linear constituent with multiplicity 1. In this paper we show that (with a few exceptions) if is a simple group or a covering group of a simple group and is an irreducible character of of degree between 32 and 100, then such a subgroup exists.

     

Security of the most significant bits of the Shamir message passing scheme

Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a ``hidden' element of a finite field of elements from rather short strings of the most significant bits of the remainder modulo of for several values of selected uniformly at random from . Unfortunately the applications to the computational security of most significant bits of private keys of some finite field exponentiation based cryptosystems given by Boneh and Venkatesan are not quite correct. For the Diffie-Hellman cryptosystem the result of Boneh and Venkatesan has been corrected and generalized in our recent paper. Here a similar analysis is given for the Shamir message passing scheme. The results depend on some bounds of exponential sums.

     

Largest known twin primes and Sophie Germain primes

The numbers are twin primes. The number is a Sophie Germain prime, i.e. and are both primes. For , the numbers , and are all primes.

     

Finding prime pairs with particular gaps

By a prime gap of size , we mean that there are primes and such that the numbers between and are all composite. It is widely believed that infinitely many prime gaps of size exist for all even integers . However, it had not previously been known whether a prime gap of size existed. The objective of this article was to be the first to find a prime gap of size , by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from to , and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form , , and their application to divisibility of binomial coefficients by a square will also be discussed.