Search Results (5)

The variance function (VF) is central to natural exponential family (NEF) theory. Prompted by an online query about whether, beyond the classical normal NEF, other real-line NEFs with bounded VFs exist, we establish three complementary sets of sufficient conditions that yield many such families. One set imposes a polynomial-growth bound on the NEF’s generating measure, ensuring rapid tail decay and a uniformly bounded VF. A second set relies on the Legendre duality, requiring a uniform positive lower bound on the second derivative of the generating function, which likewise ensures a bounded VF. The third set starts from the standard normal distribution and constructs an explicit sequence of NEFs whose Laplace transforms and VFs remain bounded. Collectively, these results reveal a remarkably broad class of NEFs whose Laplace transforms are not expressible in elementary form (apart from those stemming from the standard normal case), yet can be handled easily using modern symbolic and numerical software. Worked examples show that NEFs with bounded VFs are far more varied than previously recognized, offering practical alternatives to the normal and other classical models for real-data analysis across many fields. Full article

The rise of modern cryptographic protocols such as Zero-Knowledge proofs and secure Multi-party Computation has led to an increased demand for a new class of symmetric primitives. Unlike traditional platforms such as servers, microcontrollers, and desktop computers, these primitives are designed to be implemented in arithmetical circuits. In terms of security evaluation, arithmetization-oriented primitives are more complex compared to traditional symmetric cryptographic primitives. The arithmetization-oriented permutation Grendel employs the Legendre Symbol to increase the growth of algebraic degrees in its nonlinear layer. To analyze the security of Grendel thoroughly, it is crucial to investigate its resilience against algebraic attacks. This paper presents a preimage attack on the sponge hash function instantiated with the complete rounds of the Grendel permutation, employing algebraic methods. A technique is introduced that enables the elimination of two complete rounds of substitution permutation networks (SPN) in the sponge hash function without significant additional cost. This method can be combined with univariate root-finding techniques and Gröbner basis attacks to break the number of rounds claimed by the designers. By employing this strategy, our attack achieves a gain of two additional rounds compared to the previous state-of-the-art attack. With no compromise to its security margin, this approach deepens our understanding of the design and analysis of such cryptographic primitives. Full article

Let p be a prime and ${\mathbb{F}}_p$ be the set of integers modulo p. Let ${\chi }_p$ be a function defined on ${\mathbb{F}}_p$ such that ${\chi }_p\left(0\right)=0$ and for $a\in {\mathbb{F}}_p\backslash \left\{0\right\}$, set ${\chi }_p\left(a\right)=1$ if a is a quadratic residue modulo p and ${\chi }_p\left(a\right)=-1$ if a is a quadratic non-residue modulo p. Note that ${\chi }_p\left(a\right)=\left({\displaystyle \frac{a}{p}}\right)$ is indeed the Legendre symbol. The image of ${\chi }_p$ in the set of real numbers. In this paper, we consider the following sum ${\sum }_{x\in {\mathbb{F}}_p}{\chi }_p((x-a_1)(x-a_2)\dots (x-a_t))$ where $a_1,a_2,\dots ,a_t$ are distinct elements in ${\mathbb{F}}_p$. Full article

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels $k^{+}$ and $k^{-}$ and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences. Full article

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues $\mathrm{mod} p$ . The main methods we used are the properties of the Legendre’s symbol $\mathrm{mod} p$ , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun. Full article