The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field GF[q] and the black box can evaluate the polynomial in the field GF[qr2g,tnt+37], where n is the number of variables, then there is an algorithm to interpolate the polynomial in O(log (nt)) boolean parallel time and O(n2t log nt) processors. This algorithm yields the first efficient deterministic polynomial time algorithm (and moreover boolean NC-algorithm) for interpolating t-sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in GF[q] is

Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.

this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...

We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1) . We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of Lie type simple groups over GF (p m ) that an irreducible polynomial f of degree m over GF (p) and a primitive root of GF (p m ) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groups PSU(3; q) = 2 A 2 (q), the Suzuki groups Sz(q) = 2 B 2 (q), and the Ree groups R(q) = 2 G 2 (q). In particular, all finite groups G without composition factors of these types have presentations of length O((log jGj) 3 ). For...

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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-

The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomial-time algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the

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In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonal depth-3 circuit C(x1,..., xn) (i.e. C is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions. Our techniques generalize to the following new results: 1. Suppose we are given a depth-3 circuit (over any field F) of the form: C(x1,..., xn):= k� i=1 ℓ ei,1 i,1 · · · ℓei,s i,s where, the ℓi,j’s are linear functions living in F[x1,..., xn]. We can test whether C is zero deterministically in poly (nk, max{(1 + ei,1) · · · (1 + ei,s) | 1 � i � k}) field operations. This immediately gives a deterministic poly(nk2 d) time identity test for general depth-3 circuits of degree d. 2. We prove that if the above circuit C(x1,..., xn) computes the determinant � (or permanent) of an m × m formal matrix with a “small ” s = o then � m log m k = 2 Ω(m). Our lower bounds work for all fields F. (Previous exponential lower bounds for depth-3 only work for nonzero characteristic.) 3. We present applications of our ideas to depth-4 circuits and an exponentially faster identity test for homogeneous diagonal circuits (deterministically in poly(n k log(d)) field operations over finite fields).

. We present here some recent results on fast parallel interpolation of multivariate polynomials over finite fields. Some applications towards the general conversion algorithms for boolean functions are also formulated. Introduction We consider the general problem of interpolation of multivariate polynomials over finite fields given by black boxes (input oracles). In this setting we are given a polynomial f over GF[q], as a black box, and an information about its sparsity t (the bound on the number of nonzero coefficients). Given this, we must determine an extension GF[q s ] of GF[q], s as small as possible, and an efficient (deterministic boolean NC-algorithm, cf. [Co 85], [KR 88]) interpolation algorithm working over GF[q s ] to determine all coefficients of f in GF[q]. Such a Supported in part by Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/2-1, and by the SERC Grant GR-E 68297 general problem arises in a number of applications, e.g., in design ...