In this paper we associate to each multivariate polynomial f that is homogeneous relative to a subset of its variables a series of polynomial families Pλ(f) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in Pλ(f) is bounded from above by the circuit size of f . This provides a method for obtaining lower bounds for the circuit size of f by proving...

INTRODUCTION................................................................................... 215 I. THE HISTORY AND BACKGROUND OF 9-1-1 ............................. 218 II. THE TECHNOLOGY OF THE 9-1-1 NETWORK ........................... 223 A. Introduction............................................................................. 223 B. Background and Evolution of Our Nation’s 9-1-1 Syste...

Koiran [8] showed that if an n-variate polynomial fn of degree d (with d = n) is computed by a circuit of size s, then it is also computed by a homogeneous circuit of depth four and of size 2 √ d log(n) . Using this result, Gupta, Kamath, Kayal and Saptharishi [7] found an upper bound for the size of a depth three circuit computing fn. We improve here Koiran’s bound. Indeed, we show that it is ...

We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization: general derandomization of promise-BPP (connected to Boolean circuits), derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and derandomization of PIT over the integers ...

Secure simulations of arithmetic circuit and boolean circuit evaluations are known to save privacy while providing solutions to any probabilistic function over a field. The problem we want to solve is to select a random solution of a general combinatorial problem. Here we discuss how to specify the need of selecting a random solution of a general combinatorial problem, as a probabilistic functi...

This article describes a cost-effective method of determining the true characteristics of passive components using measurements of S-parameters. A more precise description of these component parameters can aid understanding of the arithmetic path, and the calculated values; from which equivalent circuit diagrams can be determined, both qualitatively and quantitatively, providing a basis for car...

Efficient implementation of arithmetic in finite fields is of primary importance for cryptography, coding theory, digital signal processing etc. (see, for example [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]). In this survey we consider only Boolean circuits for arithmetic operations in finite fields. Another term: bit-parallel circuits. Boolean circuits for multiplication and inversion in finite fields are...

We show that, over Q, if an n-variate polynomial of degree d = nO(1) is computable by an arithmetic circuit of size s (respectively by an algebraic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O( √ d log d log n log s)) (respectively of size exp(O( √ d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O( √ ...

For natural integer n, let Dn denote the random variable taking the values log d for d dividing n with uniform probability 1/⌧(n). Then t 7! P(Dn 6 nt) (0 6 t 6 1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the...