Pseudo-random number generators (PRNGs) play an important role to ensure the security and confidentiality of image cryptographic algorithms. Their primary function is to generate a sequence of ...numbers that possesses unpredictability and randomness, which is crucial for the algorithms to work effectively and provide the desired level of security. However, traditional PRNGs frequently encounter limitations like insufficient randomness, predictability, and vulnerability to cryptanalysis attacks. To overcome these limitations, we propose a novel method namely an elliptic curve genetic algorithm (ECGA) for the construction of an image-dependent pseudo-random number generator (IDPRNG) that merges elliptic curves (ECs) and a genetic algorithm (GA). The ECGA consists of two primary stages. First, we generate an EC-based initial sequence of random numbers using pixels of a plain-image and parameters of an EC that depart from traditional methods of population initialization. In our proposed approach, the image itself serves as the seed for the initial population in the genetic algorithm optimization, taking into account the image-dependent nature of cryptographic applications. This allows the PRNG to adapt its behavior to the unique characteristics of the input image, leading to enhanced security and improved resistance against differential attacks. Furthermore, the use of a good initial population reduces the number of generations required by a genetic algorithm which results in decreased computational cost. In the second stage, we use well-known operations of a genetic algorithm to optimize the generated sequence by maximizing a multi-variable fitness function that is based on both the information entropy and the period of the PRNG. By combining elliptic curves and genetic algorithms, we enhance the randomness and security of the ECGA. To evaluate the effectiveness and security of our generator, we conducted comprehensive experiments using various benchmark images and applied several standard tests, including the National Institute of Standards and Technology (NIST) test suite. We then compared the results with the state-of-the-art PRNGs. The experimental results demonstrate that the ECGA outperforms the state-of-the-art PRNGs in terms of uniformity, randomness, and cryptographic strength.
Here we provide three new presentations of Coxeter groups of type
A
,
B
, and
D
using prefix reversals (pancake flips) as generators. The purpose of these presentations is to advance the algebraic ...underpinnings of the pancake problem. We prove these presentations are of their respective groups by using Tietze transformations on the presentations to recover the well known presentations with generators that are adjacent transpositions. We also provide a statement for the classic pancake problem for type
D
.
The pancake graph Pn is the Cayley graph of the symmetric group Sn on n elements generated by prefix reversals. Pn has been shown to have properties that makes it a useful network scheme for parallel ...processors. For example, it is (n−1)-regular, vertex-transitive, and one can embed cycles in it of length ℓ with 6≤ℓ≤n!. The burnt pancake graph BPn, which is the Cayley graph of the group of signed permutations Bn using prefix reversals as generators, has similar properties. Indeed, BPn is n-regular and vertex-transitive. In this paper, we show that BPn has every cycle of length ℓ with 8≤ℓ≤2nn!. The proof given is a constructive one that utilizes the recursive structure of BPn.
We also present a complete characterization of all the 8-cycles in BPn for n≥2, which are the smallest cycles embeddable in BPn, by presenting their canonical forms as products of the prefix reversal generators.
The human brain is a complex network comprised of functionally and anatomically interconnected brain regions. A growing number of studies have suggested that empirical estimates of brain networks may ...be useful for discovery of biomarkers of disease and cognitive state. A prerequisite for realizing this aim, however, is that brain networks also serve as reliable markers of an individual. Here, using Human Connectome Project data, we build upon recent studies examining brain-based fingerprints of individual subjects and cognitive states based on cognitively demanding tasks that assess, for example, working memory, theory of mind, and motor function. Our approach achieves accuracy of up to 99% for both identification of the subject of an fMRI scan, and for classification of the cognitive state of a previously unseen subject in a scan. More broadly, we explore the accuracy and reliability of five different machine learning techniques on subject fingerprinting and cognitive state decoding objectives, using functional connectivity data from fMRI scans of a high number of subjects (865) across a number of cognitive states (8). These results represent an advance on existing techniques for functional connectivity-based brain fingerprinting and state decoding. Additionally, 16 different functional connectome (FC) matrix construction pipelines are compared in order to characterize the effects of different aspects of the production of FCs on the accuracy of subject and task classification, and to identify possible confounds.
The Bruhat order is a well-studied partial order on Coxeter groups and Schubert varieties. Deodhar provided several characterizations of the Bruhat order, including the so-called “
Z
-property.” ...Another partial order on Coxeter groups that is relevant in combinatorics is the absolute order, which extends the notion of the classical noncrossing partitions to Coxeter groups. In this note, we prove a result that implies that the absolute order satisfies a weaker version of the
Z
-property.
Flip Posets of Bruhat Intervals Blanco, Saúl A.
The Electronic journal of combinatorics,
10/2018, Volume:
25, Issue:
4
Journal Article
Peer reviewed
In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph $B(u,v)$ of a Bruhat interval $u,v$ into rank posets $P_{i}$ in a way that each $P_{i}$ has a unique ...maximal chain that is rising under a reflection order. In the case where each $P_{i}$ has rank three, the construction yields a combinatorial description of some terms of the complete $\textbf{cd}$-index as a sum of ordinary $\textbf{cd}$-indices of Eulerian posets obtained from each of the $P_{i}$.
In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group S(m,n), generated by ...prefix reversals. The generalized symmetric group S(m,n) is the wreath product of the cyclic group of order m and the symmetric group of order n!. Our main focus is the underlying undirected graphs, denoted by Pm(n). In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, P3(n) has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, P4(n) has all the even-length cycles. We utilize these results as base cases and show that if m>2 is even, Pm(n) has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when m>2 is odd, Pm(n) has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of Pm(n) is min{m,6} if m≥3, thus complementing the known results for m=1,2.
Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar ...characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulas exist giving the number of signed permutations that require k flips to be sorted, with 5 < k < 9.
We define the shortest path poset
SP
(
u
,
v
) of a Bruhat interval
u
,
v
, by considering the shortest
u
–
v
paths in the Bruhat graph of a Coxeter group
W
, where
u
,
v
∈
W
. We consider the case ...of
SP
(
u
,
v
) having a unique rising chain under a reflection order and show that in this case
SP
(
u
,
v
) is a Gorenstein
∗
poset. This allows us to derive the nonnegativity of certain coefficients of the complete
cd
-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.
The set of Dyck paths of length 2
n
inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution ...of two statistics for Dyck paths:
area
(the area under the path) and
rank
(the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting (
q
,
t
)-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: One refines an identity of Carlitz and Riordan; the other refines the notion of
γ
-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane’s correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the
n
-cycle (12· · ·
n
) in the absolute order on the symmetric group.
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