Skellam shrinkage: Wavelet-based intensity estimation for inhomogeneous Poisson data

The ubiquity of integrating detectors in imaging and other applications implies that a variety of real-world data are well modeled as Poisson random variables whose means are in turn proportional to an underlying vector-valued signal of interest. In this article, we first show how the so-called Skellam distribution arises from the fact that Haar wavelet and filterbank transform coefficients corresponding to measurements of this type are distributed as sums and differences of Poisson counts. We then provide two main theorems on Skellam shrinkage, one showing the near-optimality of shrinkage in the Bayesian setting and the other providing for unbiased risk estimation in a frequentist context. These results serve to yield new estimators in the Haar transform domain, including an unbiased risk estimate for shrinkage of Haar-Fisz variance-stabilized data, along with accompanying low-complexity algorithms for inference. We conclude with a simulation study demonstrating the efficacy of our Skellam shrinkage estimators both for the standard univariate wavelet test functions as well as a variety of test images taken from the image processing literature, confirming that they offer substantial performance improvements over existing alternatives.Comment: 27 pages, 8 figures, slight formatting changes; submitted for publicatio

Deterministic Chaos in Digital Cryptography

This thesis studies the application of deterministic chaos to digital cryptography. Cryptographic systems such as pseudo-random generators (PRNG), block ciphers and hash functions are regarded as a dynamic system (X, j), where X is a state space (Le. message space) and f : X -+ X is an iterated function. In both chaos theory and cryptography, the object of study is a dynamic system that performs an iterative nonlinear transformation of information in an apparently unpredictable but deterministic manner. In terms of chaos theory, the sensitivity to the initial conditions together with the mixing property ensures cryptographic confusion (statistical independence) and diffusion (uniform propagation of plaintext and key randomness into cihertext). This synergetic relationship between the properties of chaotic and cryptographic systems is considered at both the theoretical and practical levels: The theoretical background upon which this relationship is based, includes discussions on chaos, ergodicity, complexity, randomness, unpredictability and entropy. Two approaches to the finite-state implementation of chaotic systems (Le. pseudo-chaos) are considered: (i) floating-point approximation of continuous-state chaos; (ii) binary pseudo-chaos. An overview is given of chaotic systems underpinning cryptographic algorithms along with their strengths and weaknesses. Though all conventional cryposystems are considered binary pseudo-chaos, neither chaos, nor pseudo-chaos are sufficient to guarantee cryptographic strength and security. A dynamic system is said to have an analytical solution Xn = (xo) if any trajectory point Xn can be computed directly from the initial conditions Xo, without performing n iterations. A chaotic system with an analytical solution may have a unpredictable multi-valued map Xn+l = f(xn). Their floating-point approximation is studied in the context of pseudo-random generators. A cryptographic software system E-Larm ™ implementing a multistream pseudo-chaotic generator is described. Several pseudo-chaotic systems including the logistic map, sine map, tangent- and logarithm feedback maps, sawteeth and tent maps are evaluated by means of floating point computations. Two types of partitioning are used to extract pseudo-random from the floating-point state variable: (i) combining the last significant bits of the floating-point number (for nonlinear maps); and (ii) threshold partitioning (for piecewise linear maps). Multi-round iterations are produced to decrease the bit dependence and increase non-linearity. Relationships between pseudo-chaotic systems are introduced to avoid short cycles (each system influences periodically the states of other systems used in the encryption session). An evaluation of cryptographic properties of E-Larm is given using graphical plots such as state distributions, phase-space portraits, spectral density Fourier transform, approximated entropy (APEN), cycle length histogram, as well as a variety of statistical tests from the National Institute of Standards and Technology (NIST) suite. Though E-Larm passes all tests recommended by NIST, an approach based on the floating-point approximation of chaos is inefficient in terms of the quality/performance ratio (compared with existing PRNG algorithms). Also no solution is known to control short cycles. In conclusion, the role of chaos theory in cryptography is identified; disadvantages of floating-point pseudo-chaos are emphasized although binary pseudo-chaos is considered useful for cryptographic applications.Durand Technology Limite

Randomness Tests for Binary Sequences

Cryptography is vital in securing sensitive information and maintaining privacy in the today’s digital world. Though sometimes underestimated, randomness plays a key role in cryptography, generating unpredictable keys and other related material. Hence, high-quality random number generators are a crucial element in building a secure cryptographic system. In dealing with randomness, two key capabilities are essential. First, creating strong random generators, that is, systems able to produce unpredictable and statistically independent numbers. Second, constructing validation systems to verify the quality of the generators. In this dissertation, we focus on the second capability, specifically analyzing the concept of hypothesis test, a statistical inference model representing a basic tool for the statistical characterization of random processes. In the hypothesis testing framework, a central idea is the p-value, a numerical measure assigned to each sample generated from the random process under analysis, allowing to assess the plausibility of a hypothesis, usually referred to as the null hypothesis, about the random process on the basis of the observed data. P-values are determined by the probability distribution associated with the null hypothesis. In the context of random number generators, this distribution is inherently discrete but in the literature it is commonly approximated by continuous distributions for ease of handling. However, analyzing in detail the discrete setting, we show that the mentioned approximation can lead to errors. As an example, we thoroughly examine the testing strategy for random number generators proposed by the National Institute of Standards and Technology (NIST) and demonstrate some inaccuracies in the suggested approach. Motivated by this finding, we define a new simple hypothesis test as a use case to propose and validate a methodology for assessing the definition and implementation correctness of hypothesis tests. Additionally, we present an abstract analysis of the hypothesis test model, which proves valuable in providing a more accurate conceptual framework within the discrete setting. We believe that the results presented in this dissertation can contribute to a better understanding of how hypothesis tests operate in discrete cases, such as analyzing random number generators. In the demanding field of cryptography, even slight discrepancies between the expected and actual behavior of random generators can, in fact, have significant implications for data security