4,724 research outputs found
On the Affine Equivalence and Nonlinearity Preserving Bijective Mappings
It is well-known that affine equivalence relations keep nonlineaerity invariant for all Boolean functions. The set of all Boolean functions, , over \bbbf_2^n, is naturally regarded as the dimensional vector space, \bbbf_2^{2^n}. Thus, while analyzing the transformations acting on , , the group of all bijective mappings, defined from \bbbf_2^{2^n} onto itself should be considered. As it is shown in \cite{ser,ser:dog,ser:dog:2}, there exist non-affine bijective transformations that preserve nonlinearity. In this paper, first, we prove that the group of affine equivalence relations is isomorphic to the automorphism group of Sylvester Hadamard matrices. Then, we show that new nonlinearity preserving non-affine bijective mappings also exist. Moreover, we propose that the automorphism group of nonlinearity classes, should be studied as a subgroup of , since it contains transformations which are not affine equivalence relations
On Equivalence of Known Families of APN Functions in Small Dimensions
In this extended abstract, we computationally check and list the
CCZ-inequivalent APN functions from infinite families on for n
from 6 to 11. These functions are selected with simplest coefficients from
CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence
between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT
ASSOCIATION
Minimization for Generalized Boolean Formulas
The minimization problem for propositional formulas is an important
optimization problem in the second level of the polynomial hierarchy. In
general, the problem is Sigma-2-complete under Turing reductions, but
restricted versions are tractable. We study the complexity of minimization for
formulas in two established frameworks for restricted propositional logic: The
Post framework allowing arbitrarily nested formulas over a set of Boolean
connectors, and the constraint setting, allowing generalizations of CNF
formulas. In the Post case, we obtain a dichotomy result: Minimization is
solvable in polynomial time or coNP-hard. This result also applies to Boolean
circuits. For CNF formulas, we obtain new minimization algorithms for a large
class of formulas, and give strong evidence that we have covered all
polynomial-time cases
On Some Properties of Quadratic APN Functions of a Special Form
In a recent paper, it is shown that functions of the form
, where and are linear, are a good source for
construction of new infinite families of APN functions. In the present work we
study necessary and sufficient conditions for such functions to be APN
The complexity of weighted boolean #CSP*
This paper gives a dichotomy theorem for the complexity of computing the partition
function of an instance of a weighted Boolean constraint satisfaction problem. The problem
is parameterized by a finite set F of nonnegative functions that may be used to assign weights to
the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems
correspond to the special case of 0,1-valued functions. We show that computing the partition
function, i.e., the sum of the weights of all configurations, is FP#P-complete unless either (1) every
function in F is of “product type,” or (2) every function in F is “pure affine.” In the remaining cases,
computing the partition function is in P
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