arXiv subjects

Magnus Gausdal Find

Publications and source records attributed to Magnus Gausdal Find.

3 recordsLinked to original sources

arXiv2018

Multiplicative Complexity of Vector Valued Boolean Functions

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $ΣΠΣ$ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theory results, we show that functions with $n$ inputs and $n$ outputs with the highest possible nonlinearity must have at least $2.32n$ AND gates. We further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function; we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of AND gates being exactly the length of such a shortest code. Additionally we provide a function which, for general circuits, has multiplicative complexity at least $2n-3$. Finally we study the multiplicative complexity of "almost all" functions. We show that every function with $n$ bits of input and $m$ bits of output can be computed using at most $2.5(1+o(1))\sqrt{m2^n}$ AND gates.

cs.CC↗

arXiv2015

Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < ε<2$, any function with algebraic thickness $n^{3-ε}$ is constant on some affine subspace of dimension $Ω\left(n^{\fracε{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $Θ(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $Θ(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $Ω\left(2^{n^{1/6}}\right)$.

cs.CC↗

arXiv2014

On the Complexity of Computing Two Nonlinearity Measures

We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given the truth table of length $2^n$, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of $(2-ε)^{n/2}$. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit.

cs.CC↗