2,236 research outputs found
Nonlinearity and propagation characteristics of balanced boolean functions
Three of the most important criteria for cryptographically strong Boolean functions are the balancedness, the nonlinearity and the propagation criterion. The main contribution of this paper is to reveal a number of interesting properties of balancedness and nonlinearity, and to study systematic methods for constructing Boolean functions satisfying some or all of the three criteria. We show that concatenating, splitting, modifying and multiplying (in the sense of Kronecker) sequences can yield balanced Boolean functions with a very high nonlinearity. In particular, we show that balanced Boolean functions obtained by modifying and multiplying sequences achieve a nonlinearity higher than that attainable by any previously known construction method. We also present methods for constructing balanced Boolean functions that are highly nonlinear and satisfy the strict avalanche criterion (SAC). Furthermore we present methods for constructing highly nonlinear balanced Boolean functions satisfying the propagation criterion with respect to all but one or three vectors. A technique is developed to transform the vectors where the propagation criterion is not satisfied in such a way that the functions constructed satisfy the propagation criterion of high degree while preserving the balancedness and nonlinearity of the functions. The algebraic degrees of functions constructed are also discussed, together with examples illustrating the various constructions
Balanced Symmetric Functions over
Under mild conditions on , we give a lower bound on the number of
-variable balanced symmetric polynomials over finite fields , where
is a prime number. The existence of nonlinear balanced symmetric
polynomials is an immediate corollary of this bound. Furthermore, we conjecture
that are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where , and we prove various results in support of this conjecture.Comment: 21 page
Collisional dust avalanches in debris discs
We quantitatively investigate how collisional avalanches may developin debris
discs as the result of the initial break-up of a planetesimal or comet-like
object, triggering a collisional chain reaction due to outward escaping small
dust grains. We use a specifically developed numerical code that follows both
the spatial distribution of the dust grains and the evolution of their
size-frequency distribution due to collisions. We investigate how strongly
avalanche propagation depends on different parameters (e.g., amount of dust
released in the initial break-up, collisional properties of dust grains and
their distribution in the disc). Our simulations show that avalanches evolve on
timescales of ~1000 years, propagating outwards following a spiral-like
pattern, and that their amplitude exponentially depends on the number density
of dust grains in the system. We estimate a probability for witnessing an
avalanche event as a function of disc densities, for a gas-free case around an
A-type star, and find that features created by avalanche propagation can lead
to observable asymmetries for dusty systems with a beta Pictoris-like dust
content or higher. Characteristic observable features include: (i) a brightness
asymmetry of the two sides for a disc viewed edge-on, and (ii) a one-armed open
spiral or a lumpy structure in the case of face-on orientation. A possible
system in which avalanche-induced structures might have been observed is the
edge-on seen debris disc around HD32297, which displays a strong luminosity
difference between its two sides.Comment: 18 pages, 19 figures; has been accepted for publication in Astronomy
and Astrophysics, section 6. Interstellar and circumstellar matter. The
official date of acceptance is 29/08/200
A quantum algorithm to estimate the closeness to the Strict Avalanche criterion in Boolean functions
We propose a quantum algorithm (in the form of a quantum oracle) that
estimates the closeness of a given Boolean function to one that satisfies the
``strict avalanche criterion'' (SAC). This algorithm requires queries of
the Boolean function oracle, where is the number of input variables, this
is fewer than the queries required by the classical algorithm to perform the
same task. We compare our approach with other quantum algorithms that may be
used for estimating the closeness to SAC and it is shown our algorithm verifies
SAC with the fewest possible calls to quantum oracle and requires the fewest
samples for a given confidence bound
Primal-dual distance bounds of linear codes with application to cryptography
Let denote the minimum length of a linear code with
and , where is the minimum Hamming distance of and
is the minimum Hamming distance of . In this paper, we
show a lower bound and an upper bound on . Further, for small
values of and , we determine and give a generator
matrix of the optimum linear code. This problem is directly related to the
design method of cryptographic Boolean functions suggested by Kurosawa et al.Comment: 6 pages, using IEEEtran.cls. To appear in IEEE Trans. Inform. Theory,
Sept. 2006. Two authors were added in the revised versio
Construction of Balanced Boolean Functions with High Nonlinearity and Good Autocorrelation Properties
Boolean functions with high nonlinearity and good autocorrelation properties play an important role in the design of block ciphers and stream ciphers. In this paper, we give a method to construct balanced Boolean functions on variables, where is an even integer, satisfying strict avalanche criterion (SAC). Compared with the known balanced Boolean functions with SAC property, the constructed functions possess the highest nonlinearity and the best global avalanche characteristics (GAC) property
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