Communication complexity of some number theoretic functions

Complexity Theoretic Aspects of Some Cryptographic Functions

In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from {−1, 1} . It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N, where ε > 0 denotes a fixed constant. For instance, the size of a two-layer threshold circuit (with p...

Some Number Theoretic Results

This note consists of two rather separate topics . In §1 we generalize a topic from combinatorial number theory to get an order of magnitude for the number of elements in a maximal set of residues (mod p) such that sums of different numbers of elements from this set are distinct. We show that the correct order is ep'" 3 although we are unable to establish the correct value for the constant c . ...

On Some Communication Complexity Problems Related to THreshold Functions

We study the computation of threshold functions using formulas over the basis fAND, OR, NOTg, with the aim of unifying the lower bounds of Hansel, Krichevskii, and Khrapchenko. For this we consider communication complexity problems related to threshold function computation. We obtain an upper bound for a communication problem used by Karchmer and Wigderson to derive the depth version of the Khr...

Open problems in number theoretic complexity, II

(non)automaticity of Number Theoretic Functions

Denote by λ(n) Liouville’s function concerning the parity of the number of prime divisors of n. Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that λ(n) is not k–automatic for any k > 2. This yields that P ∞ n=1 λ(n)X ∈ Fp[[X]] is transcendental over Fp(X) for any prime p > 2. Similar results are...