We define a new structured and general model of computation: circuits using arbitrary fan-in arithmetic gates over the characteristic-two finite fields (F,:,). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related, then the arithmetic circuit depth and t...

We present a novel and efficient, in terms of circuit depth, design for Shor’s quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper’s adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively archit...

We present an Attribute Based Encryption system where access policies are expressed as polynomial size arithmetic circuits. We prove security against arbitrary collusions of users based on the learning with errors problem on integer lattices. The system has two additional useful properties: first, it naturally handles arithmetic circuits with arbitrary fan-in (and fan-out) gates. Second, secret...

Many dynamic programming algorithms for discrete 0-1 optimization problems are just special (recursively constructed) tropical (min,+) or (max,+) circuits. A problem is homogeneous if all its feasible solutions have the same number of 1s. Jerrum and Snir [JACM 29 (1982), pp. 874– 897] proved that tropical circuit complexity of homogeneous problems coincides with the monotone arithmetic circuit ...

Matrix rigidity is a notion put forth by Valiant [Val77] as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant’s plan has been found. Recently, Alman and Williams [AW17] showed, contrary to common belief, that the 2 × 2 Hadamard matrix...

FPGAs have advantage over ASICs such as rapid prototyping, in circuit programmability, lower NRE costs and results in more economical designs. In conventional pipelining technique, the operating frequency is increased by dividing the combinational logic into number of stages and registers are introduced between the stages. All the registers are fed with a global clock. This improves the speed o...

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial sλ(x1, . . . , xk) labeled by a partition λ = (λ1 ≥ λ2 ≥ · · · ) is bounded by O(log(λ1)) provided the number of variables k is fixed.

This paper demonstrates the ability of genetic programming to evolve analog circuits that perform digital functions and mixed analog-digital circuits. The evolved circuits include two purely digital circuits (a 100 nano-second NAND circuit and a two-instruction arithmetic logic unit circuit) and one mixed-signal circuit, namely a three-input digital-to-analog converter.

A rational number can be naturally presented by an arithmetic computation (AC): a sequence of elementary arithmetic operations starting from a fixed constant, say 1. The asymptotic complexity issues of such a representation are studied e.g. in [2, 9] in the framework of the algebraic complexity theory over arbitrary field. Here we study a related problem of the complexity of performing arithmet...