Integer Addition and Hamming Weight

TL;DR

This paper investigates how adding a fixed integer with a complex binary structure affects the Hamming weight distribution of numbers, revealing implications for computational complexity and circuit size requirements.

Contribution

It demonstrates that addition by such an integer significantly increases the Hamming weight of many low-weight integers, impacting the complexity of powering maps in circuit models.

Findings

Addition by lpha with lpha many blocks causes many low Hamming weight integers to become high Hamming weight.

Powering by lpha with many blocks requires exponential-size, bounded-depth circuits over _2.

Results have applications in complexity theory, particularly in circuit complexity lower bounds.

Abstract

We study the effect of addition on the Hamming weight of a positive integer. Consider the first $2^n$ positive integers, and fix an $\alpha$ among them. We show that if the binary representation of $\alpha$ consists of $\Theta(n)$ blocks of zeros and ones, then addition by $\alpha$ causes a constant fraction of low Hamming weight integers to become high Hamming weight integers. This result has applications in complexity theory to the hardness of computing powering maps using bounded-depth arithmetic circuits over $\mathbb{F}_2$. Our result implies that powering by $\alpha$ composed of many blocks require exponential-size, bounded-depth arithmetic circuits over $\mathbb{F}_2$.