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Private set intersection (PSI) is a cryptographic primitive that allows two or more parties to learn the intersection of their input sets and nothing else. In this paper, we introduce a new secure multi-party quantum computation for the greatest common divisor (GCD) and a PSI protocol based on it. The protocol extends the recent least common multiple (LCM)-based quantum private set union protocol by Liu et al. Performance analysis verifies the correctness and demonstrates the entire security of the suggested protocols in the semi-honest model. Additionally, it has been demonstrated that the complexity is efficient in terms of the input set size.

In this paper, we define a Jordan totient graph as an undirected graph with a vertex set $V$ consisting of first $q^2-1$ natural numbers, where $q$ is an odd prime, and an edge set $E$ consisting of a pair of distinct nodes with the property that these two along with n have no common divisor greater than one. Our investigation focuses on three primary objectives. First, we examine the applicability of standard graph-theoretic concepts to Jordan totient graphs, specifically exploring instances where the graph parameters $k$ and $n=q^2$ are derived from the Jordan totient function $J_k(n)$. Second, we develop a combinatorial formula to enumerate the number of triangles within these graphs for given $k$ and $n$. Finally, we propose a model for scalable free social networks using Jordan totient graphs, differentiating between two groups of entities: one with internal isolation but external connections, and the other with unrestricted relationships.

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

There is a classical correspondence between edge-signed plane graphs and link diagrams. Determining component number of links corresponding to plane graphs may be one of the first problems in studying links by using graphs. There has been several early studies in this aspect, for example, the component number of links formed from 2-dimensional square lattices (4⁴) has been determined. In this paper, we determine the component number of links corresponding to 2-dimensional triangular (3⁶) and honeycomb (6³) lattices with free or cyclic boundary condition.

This paper deals with triangular norms and conorms defined on the extended set N of natural numbers ordered by divisibility. From the fundamental theorem of arithmetic, N can be identified with a lattice of functions from the set of primes to the complete chain {0, 1, 2, …, +∞}, thus our knowledge about (divisible) t-norms on this chain can be applied to the study of t-norms on N. A characterization of those t-norms on N which are a direct product of t-norms on {0, 1, 2, …, +∞} is given and, after introducing the concept of T-prime (prime with respect to a t-norm T), a theorem about the existence of a T-prime decomposition is obtained. This result generalizes the fundamental theorem of arithmetic.

A (commutative unital) ring is said to have FSP if it has only finitely many unital subrings. The singly generated rings that have FSP have been classified. Thus, a characterization of the rings satisfying FSP is obtained by proving that a ring R has FSP if and only if either R is finite or R = ℤ[t₁, …, t_n] ⊇ ℤ where ℤ[t_i] has FSP for each i = 1, …, n. Also, the following characterization is given for the nontrivial ring direct products Π_{i ∈ I} R_i that have FSP: I is finite, each R_i has FSP, and there is at most one i ∈ I such that R_i has characteristic 0.

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(gcd(f(n),g(n)))$ over all integers $n$. It is known that $S\le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $p{s}^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.

The classical Menon’s identity [P. K. Menon, On the sum $\sum (a-1,n)[(a,n)=1]$, J. Indian Math. Soc.$($N.S.$)$29 (1965) 155–163] states that

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd’s and lcm’s of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordellès [Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Seq. 10 (2007) Article ID:07.9.2, 13pp], and extend certain related results of Hilberdink and Tóth [On the average value of the least common multiple of $k$ positive integers, J. Number Theory 169 (2016) 327–341]. We also formulate some conjectures and open problems.