Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

Abstract. Given a “black box ” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shiftα∈Q, the exponents 0≤e1< e2<···<et, and the coefficients c1,...,ct∈Q\{0} such that f (x)=c1(x−α) e1 + c2(x−α) e2 +···+ct(x−α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size and in particular is logarithmic in deg f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information. We give both an unconditional deterministic algorithm which is polynomial-time but has a rather high complexity, and a more practical probabilistic algorithm which relies on some unknown constants.

We say that a bipartite graph Γ(V1 ∪ V2, E) has bi-degree r, s if every vertex from V1 has degree r and every vertex from V2 has degree s. Γ is called an (r, s, t)–graph if, additionally, the girth of Γ is 2t. For t> 3, very few examples of (r, s, t)–graphs were previously known. In this paper we give a recursive construction of (r, s, t)–graphs for all r, s, t ≥ 2, as well as an algebraic construction of such graphs for all r, s ≥ t ≥ 3.

Abstract. For any C ∈ [0, ∞] a compact group automorphism T: X → X is constructed with the property that 1 n log |{x ∈ X | T n (x) = x} | − → C. This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy. 1.

We study the average multiplicative order of elements modulo n and show that its behaviour is very close to the behaviour of the largest possible multiplicative order of elements modulo n given by the Carmichael function #(n). 2000 Mathematics Subject Classification: Primary 11N37, 11N64; Secondary 20K01 1

For a given residue class a (mod m) with gcd(a, m) = 1, upper bounds are obtained on the smallest value of n with ϕ(n) ≡ a (mod m). Here, as usual ϕ(n) denotes the Euler function. These bounds complement a result of W. Narkiewicz on the asymptotic uniformity of distribution of values of the Euler function in reduced residue classes modulo m. Some discussion and results are also given for classes with gcd(a, m)>1, in which case such n do not always exist, and also on the related problem for ‘cototients’.

We show that the new pseudo-random number generator, introduced recently by M. Naor and O. Reingold, possess one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions.

Abstract. Let Lq(s) be the product of Dirichlet L-functions modulo q. Then Lq(s) has at most one zero in the region

When using Built-In Self Test (BIST) for testing VLSI circuits the circuit response to an input test sequence, which may consist of thousands to millions of bits, is compacted into a signature which consists of only tens of bits. Usually a linear feedback shift register (LFSR) is used for response compaction via polynomial division. The compacting function is a many-to-one function and as a result some erroneous responses may be mapped to the same signature as the good response. This is known as aliasing. In this paper we deal with the selection of a feedback polynomial for the compacting LFSR, such that an erroneous response resulting from any modeled fault is mapped to a signature that is different from that for the good response. Such LFSRs are called zero-aliasing LFSRs. Only zero-aliasing LFSRs with primitive or irreducible feedback polynomials are considered due to their suitability for BIST test pattern generation. Upper bounds are derived for the least degree irreducible and ...

We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two N-bit integers that improves the O(N · log N · log log N) algorithm by Schönhage-Strassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer’s algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürer’s algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar. 1