. This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the split-radix FFT trick, Good's trick, the SchonhageStrassen trick, Schonhage's trick, Nussbaumer's trick, the cyclic SchonhageStrassen trick, and the Cantor-Kaltofen theorem. It emphasizes the underlying ring homomorphisms. 1.

Let S0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f1,..., fp. This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S0. For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S0. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f1,..., fp and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f1,..., fp.

This survey explains how some useful arithmetic operations can be sped up from quadratic time to essentially linear time.

Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.

. The multiplicative complexity c(f) of a Boolean function f is the minimum number of AND gates in a circuit representing f which employs only AND, XOR and NOT gates. A constructive upper bound, c(f) = 2 n 2 +1 \Gamma n=2 \Gamma 2, for any Boolean function f on n variables (n even) is given. A counting argument gives a lower bound of c(f) = 2 n 2 \Gamma O(n). Thus we have shown a separation, by an exponential factor, between worst-case Boolean complexity (which is known to be \Theta(2 n n \Gamma1 )) and worst-case multiplicative complexity. A construction of circuits for symmetric Boolean functions on n variables, requiring less than n + 3 p n AND gates, is described. 1 Introduction. A fair amount of research in Boolean circuit complexity is devoted to the following problem: Given a Boolean function and a supply of gates that perform certain basic operations, construct a circuit which corresponds (in some way) to the function and is optimal (in some sense). A well-studied...

Abstract. Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g 3+ɛ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g 2.376). This represents a significant improvement over the previous record of O(g 4) field operations (also after a precomputation) for general curves of genus g. 1.

One of the leading problems of algebraic complexity theory is matrix multiplication. The naïve multiplication of two n × n matrices uses n 3 multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and

Abstract. Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent. 1.

We report on a joint project with Joe Buhler, Richard Crandall, Reijo Ernvall, and Tauno Metsankyla dealing with the computation of irregular primes and cyclotomic invariants for primes between four and eight million. This extends previous computations of Buhler et al. [4]. Our computation of the irregular primes is based on a new approach which has originated in the study of Stickelberger codes [13]. It reduces the problem to that of finding zeros of a polynomial over F p of degree ! (p \Gamma 1)=2 among the quadratic residues. Use of fast polynomial gcd-algorithms gives an O(p log 2 p log log p)-algorithm for this task. By employing the Schonhage-Strassen algorithm for fast integer multiplication combined with a version of fast multiple evaluation of polynomials we design an algorithm with running time O(p log p log log p). This algorithm is particularly efficient when run on primes p for which p \Gamma 1 has small prime factors. We also give some improvements on the previous imple...

Abstract. This paper presents a simplified list-decoding algorithm to correct any number w of errors in any alternant code of any length n with any designed distance t + 1 over any finite field Fq; in particular, in the classical Goppa codes used in the McEliece and Niederreiter public-key cryptosystems. The algorithm is efficient for w close to, and in many cases slightly beyond, the Fq Johnson bound J ′ = n ′ − √ n ′ (n ′ − t − 1) where n ′ = n(q − 1)/q, assuming t + 1 ≤ n ′. In the typical case that qn/t ∈ (lg n) O(1) and that the parent field has (lg n) O(1) bits, the algorithm uses n(lg n) O(1) bit operations for w ≤ J ′ − n/(lg n) O(1) ; O(n 4.5) bit operations for w ≤ J ′ + o((lg n) / lg lg n); and n O(1) bit operations for w ≤ J ′ + O((lg n) / lg lg n). 1