Integer Addition and Hamming Weight

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 ${\mathbb{F}}_2$.

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References in corpus (1)

• Represent MOD function by low degree polynomial with unbounded one-sided error