Random Integer Lattice Generation via the Hermite Normal Form

Lattices used in cryptography are integer lattices. Defining and generating a “random integer lattice” are interesting topics. A generation algorithm for a random integer lattice can be used to serve as a random input of all the lattice algorithms. In this paper, we recall the definition of the random integer lattice given by G. Hu et al. and present an improved generation algorithm for it via the Hermite normal form. It can be proven that with probability ≥0.99, this algorithm outputs an n-dim random integer lattice within O(n2) operations.

Joint Poisson distribution of prime factors in sets

Abstarct Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

Combinatorial Models of the Distribution of Prime Numbers

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

Combinatorial Models of the Distribution of Prime Numbers

This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.

Joint optimization strategy for QoE-aware encrypted video caching and content distributing in multi-edge collaborative computing environment

The video request service of users in 5G network will explode, and adaptive bit rate technology can provide users with reliable video response. Placing video resources on edge servers close to users can overcome the problem of excessive network load similar to traditional centralized cloud platform solutions. Moreover, multiple edge servers can provide caching and transcoding support by collaboration mechanisms, which further improves users’ Quality of Experience (QoE). However, the design difficulty of video caching and content distribution strategies is increased due to the diversity of collaboration mechanisms and the competition between local and collaborative services of edge servers for computing and storage resources. In order to solve this problem, video cache and content distribution problem is modeled as random integer programming problem in the multi-edge server at most two-hop collaboration scenario. In order to improve the security of video data transmission, the video stream is encrypted using an encryption algorithm based on Logistic chaotic-Quantum-dot Cellular Automata (QCA). For improving the efficiency of solving integer programming problems, this paper uses a pyramid intelligent evolution algorithm based on optimal cooperation strategy to solve this problem. Simulation experiments show that our proposed method can obtain higher QoE value compared with several newer methods. In addition, the average access delay of proposed method is shortened by more than 27.98%, which verifies its reliability.

The Smith normal form distribution of a random integer matrix

International audience We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.

On the largest part size and its multiplicity of a random integer partition

Let λ be a partition of the positive integer n chosen uniformly at random among all such partitions. Let Ln = Ln(λ) and Mn = Mn(λ) be the largest part size and its multiplicity, respectively. For large n, we focus on a comparison between the partition statistics Ln and LnMn. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of LnMn – Ln grows as fast as {1 \over 2}\log n . We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.