154 research outputs found
On the Design of Cryptographic Primitives
The main objective of this work is twofold. On the one hand, it gives a brief
overview of the area of two-party cryptographic protocols. On the other hand,
it proposes new schemes and guidelines for improving the practice of robust
protocol design. In order to achieve such a double goal, a tour through the
descriptions of the two main cryptographic primitives is carried out. Within
this survey, some of the most representative algorithms based on the Theory of
Finite Fields are provided and new general schemes and specific algorithms
based on Graph Theory are proposed
On Self-Dual Quantum Codes, Graphs, and Boolean Functions
A short introduction to quantum error correction is given, and it is shown
that zero-dimensional quantum codes can be represented as self-dual additive
codes over GF(4) and also as graphs. We show that graphs representing several
such codes with high minimum distance can be described as nested regular graphs
having minimum regular vertex degree and containing long cycles. Two graphs
correspond to equivalent quantum codes if they are related by a sequence of
local complementations. We use this operation to generate orbits of graphs, and
thus classify all inequivalent self-dual additive codes over GF(4) of length up
to 12, where previously only all codes of length up to 9 were known. We show
that these codes can be interpreted as quadratic Boolean functions, and we
define non-quadratic quantum codes, corresponding to Boolean functions of
higher degree. We look at various cryptographic properties of Boolean
functions, in particular the propagation criteria. The new aperiodic
propagation criterion (APC) and the APC distance are then defined. We show that
the distance of a zero-dimensional quantum code is equal to the APC distance of
the corresponding Boolean function. Orbits of Boolean functions with respect to
the {I,H,N}^n transform set are generated. We also study the peak-to-average
power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove
that PAR_IHN of a quadratic Boolean function is related to the size of the
maximum independent set over the corresponding orbit of graphs. A construction
technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It
is finally shown that both PAR_IHN and APC distance can be interpreted as
partial entanglement measures.Comment: Master's thesis. 105 pages, 33 figure
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
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