We consider the asymptotic behavior of a probability density function for the sum of any two lognormally distributed random variables that are nontrivially correlated. We show that both the left and right tails can be approximated by some simple functions. Furthermore, the same techniques are applied to determine the tail probability density function for a ratio statistic, and for a sum with more than two lognormally distributed random variables under some stricter conditions. The results yield new insights into the problem of characterization for a sum of lognormally distributed random variables and demonstrate that there is a need to revisit many existing approximation methods.

To provide secure communication, the authentication-and-key-agreement scheme plays a vital role in multi-server environments, Internet of Things (IoT), wireless sensor networks (WSNs), etc. This scheme enables users and servers to negotiate for a common session initiation key. Our proposal first analyzes Amin et al.’s authentication scheme based on RSA and proves that it cannot provide perfect forward secrecy and user un-traceability, and is susceptible to offline password guessing attack and key-compromise user impersonation attack. Secondly, we provide that Srinivas et al.’s multi-server authentication scheme is not secured against offline password guessing attack and key-compromise user impersonation attack, and is unable to ensure user un-traceability. To remedy such limitations and improve computational efficiency, we present a multi-server two-factor authentication scheme using elliptic curve cryptography (ECC). Subsequently, employing heuristic analysis and Burrows–Abadi–Needham logic (BAN-Logic) proof, it is proven that the presented scheme provides security against all known attacks, and in particular provides user un-traceability and perfect forward security. Finally, appropriate comparisons with prevalent works demonstrate the robustness and feasibility of the presented solution in multi-server environments.

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Let a be a maximum period feedback with carry shift register sequence (l-sequence) with connection integer q = p e and period T = p e−1 (p − 1). It is shown that the expected value of its autocorrelations is 0, and its variance is O(q ln 4 q). Thus when q is sufficiently large, with high probability, the autocorrelations are low. Furthermore, it is shown that when e ≥ 2, for any integer i, 1 ≤ i ≤ e/2, when the shift is a multiple of T /2p i , the absolute value of the autocorrelations of a is T /p 2i−1 , and the sign relies on the parity of the multiple. Introduction.Pseudorandom sequences are important in many areas of communications and computing such as cryptography, spread spectrum communications, error correcting codes, and quasi-Monte Carlo integration. In the study of pseudorandom sequences, we are often interested in the correlation properties of the sequences. These properties not only are important measures of randomness [4] but also have practical applications in spread spectrum communication systems, radar systems, cryptanalysis, and so on.Feedback with carry shift register (FCSR) sequences, especially the l-sequences, have many fine pseudorandom properties analogous to those of m-sequences. Let q = p e , with p an odd prime integer and e ≥ 1, and let 2 be a primitive root modulo q. The class of binary sequences known as l-sequences can be described in several ways [8], [9]. An l-sequence is the output sequence from a maximum period FCSR with connection number q. It is the 2-adic expansions of a rational number r/q, where gcd(r, q) = 1, and it is the sequence a n = (A · 2 −n (mod q))(mod 2), where gcd(A, q) = 1.Up to now, research has been done on the distribution properties and linear complexity of l-sequences [5], [6], [7], [8], [9], [13], [14]. The lattice test on such sequences has also been done by Couture and L'Ecuyer [1], [2], [3] and L'Ecuyer [10], [11]. Their ordinary autocorrelations have not been studied, although their arithmetic autocorrelations have been shown to be zero [5].The autocorrelation function of a binary periodic sequence a = (a 0 , a 1 , a 2 , . . . ) with period T is defined as C a (τ ) =

Let a be an l-sequence generated by a feedback-with-carry shift register with connection integer q = p e , where p is an odd prime and e ≥ 1. Goresky and Klapper conjectured that when p e / ∈ {5, 9, 11, 13}, all decimations of a are cyclically distinct. When e = 1 and p > 13, they showed that the set of distinct decimations is large and, in some cases, all deciamtions are distinct. In this article, we further show that when e ≥ 2 and p e = 9, all decimations of a are also cyclically distinct. Index TermsFeedback-with-carry shift register (FCSR), l-sequences, arithmetic correlations, 2-adic numbers, integer residue ring, primitive sequences.