3 research outputs found
Nonlinearity Computation for Sparse Boolean Functions
An algorithm for computing the nonlinearity of a Boolean function from its
algebraic normal form (ANF) is proposed. By generalizing the expression of the
weight of a Boolean function in terms of its ANF coefficients, a formulation of
the distances to linear functions is obtained. The special structure of these
distances can be exploited to reduce the task of nonlinearity computation to
solving an associated binary integer programming problem. The proposed
algorithm can be used in cases where applying the Fast Walsh transform is
infeasible, typically when the number of input variables exceeds 40
Nonlinearity of Boolean functions: an algorithmic approach based on multivariate polynomials
We compute the nonlinearity of Boolean functions with Groebner basis
techniques, providing two algorithms: one over the binary field and the other
over the rationals. We also estimate their complexity. Then we show how to
improve our rational algorithm, arriving at a worst-case complexity of
operations over the integers, that is, sums and doublings. This way,
with a different approach, we reach the same complexity of established
algorithms, such as those based on the fast Walsh transform.Comment: arXiv admin note: text overlap with arXiv:1404.247
Quantum and Randomised Algorithms for Non-linearity Estimation
Non-linearity of a Boolean function indicates how far it is from any linear
function. Despite there being several strong results about identifying a linear
function and distinguishing one from a sufficiently non-linear function, we
found a surprising lack of work on computing the non-linearity of a function.
The non-linearity is related to the Walsh coefficient with the largest absolute
value; however, the naive attempt of picking the maximum after constructing a
Walsh spectrum requires queries to an -bit function. We
improve the scenario by designing highly efficient quantum and randomised
algorithms to approximate the non-linearity allowing additive error, denoted
, with query complexities that depend polynomially on . We
prove lower bounds to show that these are not very far from the optimal ones.
The number of queries made by our randomised algorithm is linear in ,
already an exponential improvement, and the number of queries made by our
quantum algorithm is surprisingly independent of . Our randomised algorithm
uses a Goldreich-Levin style of navigating all Walsh coefficients and our
quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude
amplification and amplitude estimation to improve upon the existing quantum
versions of the Goldreich-Levin technique.Comment: Accepted in ACM Transactions on Quantum Computin