767 research outputs found
Invariants for EA- and CCZ-equivalence of APN and AB functions
An (n,m)-function is a mapping from to . Such functions have numerous applications across mathematics and computer science, and in particular are used as building blocks of block ciphers in symmetric cryptography. The classes of APN and AB functions have been identified as cryptographically optimal with respect to the resistance against two of the most powerful known cryptanalytic attacks, namely differential and linear cryptanalysis. The classes of APN and AB functions are directly related to optimal objects in many other branches of mathematics, and have been a subject of intense study since at least the early 90’s. Finding new constructions of these functions is hard; one of the most significant practical issues is that any tentatively new function must be proven inequivalent to all the known ones. Testing equivalence can be significantly simplified by computing invariants, i.e. properties that are preserved by the respective equivalence relation. In this paper, we survey the known invariants for CCZ- and EA-equivalence, with a particular focus on their utility in distinguishing between inequivalent instances of APN and AB functions. We evaluate each invariant with respect to how easy it is to implement in practice, how efficiently it can be calculated on a computer, and how well it can distinguish between distinct EA- and CCZ-equivalence classes.publishedVersio
The complexity of Boolean functions from cryptographic viewpoint
Cryptographic Boolean functions must be complex to satisfy Shannon\u27s principle of confusion. But the cryptographic viewpoint on complexity is not the same as in circuit complexity.
The two main criteria evaluating the cryptographic complexity of Boolean functions on are the nonlinearity (and more generally the -th order nonlinearity, for every positive ) and the algebraic degree. Two other criteria have also been considered: the algebraic thickness and the non-normality. After recalling the definitions of these criteria and why, asymptotically, almost all Boolean functions are deeply non-normal and have high algebraic degrees, high (-th order) nonlinearities and high algebraic thicknesses, we study the relationship between the -th order nonlinearity and a recent cryptographic criterion called the algebraic immunity. This relationship strengthens the reasons why the algebraic immunity can be considered as a further cryptographic complexity criterion
Synchronization of spatiotemporal semiconductor lasers and its application in color image encryption
Optical chaos is a topic of current research characterized by
high-dimensional nonlinearity which is attributed to the delay-induced
dynamics, high bandwidth and easy modular implementation of optical feedback.
In light of these facts, which adds enough confusion and diffusion properties
for secure communications, we explore the synchronization phenomena in
spatiotemporal semiconductor laser systems. The novel system is used in a
two-phase colored image encryption process. The high-dimensional chaotic
attractor generated by the system produces a completely randomized chaotic time
series, which is ideal in the secure encoding of messages. The scheme thus
illustrated is a two-phase encryption method, which provides sufficiently high
confusion and diffusion properties of chaotic cryptosystem employed with unique
data sets of processed chaotic sequences. In this novel method of cryptography,
the chaotic phase masks are represented as images using the chaotic sequences
as the elements of the image. The scheme drastically permutes the positions of
the picture elements. The next additional layer of security further alters the
statistical information of the original image to a great extent along the
three-color planes. The intermediate results during encryption demonstrate the
infeasibility for an unauthorized user to decipher the cipher image. Exhaustive
statistical tests conducted validate that the scheme is robust against noise
and resistant to common attacks due to the double shield of encryption and the
infinite dimensionality of the relevant system of partial differential
equations.Comment: 20 pages, 11 figures; Article in press, Optics Communications (2011
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Multimode Spontaneous Parametric Down-Conversion in the Lossy Medium
We study the process of multimode Spontaneous Parametric Down--Conversion
(SPDC) in the lossy, one dimensional waveguide. We propose a description using
first order Correlation Functions (CF) in the fluorescence fields, as a very
fruitful and easy approach providing us with a complete information about the
final multimode state. We formulate the equation of the evolution of the
multimode CF along the crystal using four characteristic length scales. We
solve it analytically in the one mode case and numerically in the multimode
case. We capture simultaneous effects of three wave mixing with ultrashort
pump, linear propagation and attenuation, and we are able to divide the
evolution into three stages and predict it qualitatively. We find that losses
do not destroy the quantum properties of SPDC but stabilize the final state
Towards a deeper understanding of APN functions and related longstanding problems
This dissertation is dedicated to the properties, construction and analysis of APN and AB functions. Being cryptographically optimal, these functions lack any general structure or patterns, which makes their study very challenging. Despite intense work since at least the early 90's, many important questions and conjectures in the area remain open. We present several new results, many of which are directly related to important longstanding open problems; we resolve some of these problems, and make significant progress towards the resolution of others.
More concretely, our research concerns the following open problems: i) the maximum algebraic degree of an APN function, and the Hamming distance between APN functions (open since 1998); ii) the classification of APN and AB functions up to CCZ-equivalence (an ongoing problem since the introduction of APN functions, and one of the main directions of research in the area); iii) the extension of the APN binomial over into an infinite family (open since 2006); iv) the Walsh spectrum of the Dobbertin function (open since 2001); v) the existence of monomial APN functions CCZ-inequivalent to ones from the known families (open since 2001); vi) the problem of efficiently and reliably testing EA- and CCZ-equivalence (ongoing, and open since the introduction of APN functions).
In the course of investigating these problems, we obtain i.a. the following results: 1) a new infinite family of APN quadrinomials (which includes the binomial over ); 2) two new invariants, one under EA-equivalence, and one under CCZ-equivalence; 3) an efficient and easily parallelizable algorithm for computationally testing EA-equivalence; 4) an efficiently computable lower bound on the Hamming distance between a given APN function and any other APN function; 5) a classification of all quadratic APN polynomials with binary coefficients over for ; 6) a construction allowing the CCZ-equivalence class of one monomial APN function to be obtained from that of another; 7) a conjecture giving the exact form of the Walsh spectrum of the Dobbertin power functions; 8) a generalization of an infinite family of APN functions to a family of functions with a two-valued differential spectrum, and an example showing that this Gold-like behavior does not occur for infinite families of quadratic APN functions in general; 9) a new class of functions (the so-called partially APN functions) defined by relaxing the definition of the APN property, and several constructions and non-existence results related to them.Doktorgradsavhandlin
Experimental investigation of pulsed entangled photons and photonic quantum channels
The development of key devices and systems in quantum information technology,
such as entangled particle sources, quantum gates and quantum cryptographic
systems, requires a reliable and well-established method for characterizing how
well the devices or systems work. We report our recent work on experimental
characterization of pulsed entangled photonic states and photonic quantum
channels, using the methods of state and process tomography. By using state
tomography, we could reliably evaluate the states generated from a two-photon
source under development and develop a highly entangled pulsed photon source.
We are also devoted to characterization of single-qubit and two-qubit photonic
quantum channels. Characterization of typical single-qubit decoherence channels
has been demonstrated using process tomography. Characterization of two-qubit
channels, such as classically correlated channels and quantum mechanically
correlated channels is under investigation. These characterization techniques
for quantum states and quantum processes will be useful for developing photonic
quantum devices and for improving their performances.Comment: 12 pages, 8 figures, in Quantum Optics in Computing and
Communications, Songhao Liu, Guangcan Guo, Hoi-Kwong Lo, Nobuyuki Imoto,
Eds., Proceedings of SPIE Vol. 4917, pp.13-24 (2002
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