We study the effect of addition on the Hamming weight of a positive integer. Consider the first @math positive integers, and fix an @math among them. We show that if the binary representation of @math consists of @math blocks of zeros and ones, then addition by @math 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 @math . Our result implies that powering by @math composed of many blocks require exponential-size, bounded-depth arithmetic circuits over @math .
In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by @math or @math in the polynomial ring @math , @math are coprime, which is called over @math . The immunity of @math is lower bounded by @math , which is achievable when @math is a multiple of @math ; the immunity of @math is exactly @math for every @math and @math . Our result improves the previous bound @math by Green. We observe how immunity over @math is related to @math circuit lower bound. For example, if the immunity of @math over @math is lower bounded by @math , and @math , then @math requires @math circuit of exponential size to compute. We study the complexity of computing the kth-power of an element of F2n by constant depth arithmetic circuits over F2 (also known as ACP). Our study encompasses the complexity of basic arithmetic operations such as computing cube-root and computing cubic-residuosity of elements of F2n. Our main result is that these problems require exponential size circuits. We also derive strong average-case versions of these results. For example, we show that no subexponential-size, constant-depth, arithmetic circuit over F2 can correctly compute the cubic residue symbol for more than 1 3 + o(1) fraction of the elements of F2n. As a corollary, we deduce a character sum bound showing that the cubic residue character over F2n is uncorrelated with all degree-d n-variate F2 polynomials (viewed as functions over F2n in a natural way), provided d l ne for some universal e > 0. Classical methods (based on van der Corput differencing and the Weil bounds) show this only for d l log(n). Our proof revisits the classical Razborov-Smolensky method for circuit lower bounds, and executes an analogue of it in the land of univariate polynomials over F2n. The tools we use come from both F2n and F2n. In recent years, this interplay between F2n and F2n has played an important role in many results in pseudorandomness, property testing and coding theory.