Narrow Results

The Global Avalanche Characteristics (including the sum-of-squares indicator and the absolute indicator) measure the overall avalanche characteristics of a cryptographic Boolean function. Son et al. (1998) gave the lower bound on the sum-of-squares indicator for a balanced Boolean function. In this paper, we give a sufficient and necessary condition on a balanced Boolean function reaching the lower bound on the sum-of-squares indicator. We also analyze whether these balanced Boolean functions exist, and if they reach the lower bounds on the sum-of-squares indicator or not. Our result implies that there does not exist a balanced Boolean function with n-variable for odd n(n ≥ 5). We conclude that there does not exist a m(m ≥ 1)-resilient function reaching the lower bound on the sum-of-squares indicator with n-variable for n ≥ 7.

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of , we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behaves well against fast algebraic attacks.

In this paper, we present certain algorithms for clustering the vertices of fuzzy graphs(FGs) and intuitionistic fuzzy graphs(IFGs). These algorithms are based on the edge density of the given graph. We apply the algorithms to practical problems to derive the most prominent cluster among them. We also introduce parameters for intuitionistic fuzzy graphs.

We define the concepts of balanced set and absorbing set in modules over topological rings, which coincide with the usual concepts when restricting to topological vector spaces. We show that in a topological module over an absolute semi-valued ring whose invertibles approach $0$, every neighborhood of $0$ is absorbing. We also introduce the concept of total closed unit neighborhood of zero (total closed unit) and prove that the only total closed unit of the quaternions $ℍ$ is its closed unit ball $B_{ℍ}$. On the other hand, we also prove that if $A$ is an absolute semi-valued unital real algebra, then its closed unit ball $B_A$ is a total closed unit. Finally, we study the geometry of modules via the extreme points and the internal points, showing that no internal point can be an extreme point and that absorbance is equivalent to having $0$ as an internal point.

Consider a synchronous system of clients and servers whereby client requests may be satisfied by any one of the servers the clients are assigned to and where the association between clients and servers is determined by an arbitrary but predefined apportionment of the set of servers to clients and of clients to servers. A client can get the required service from any of the servers it is assigned to and at each time step all clients make a request to the servers which then provide the service according to requests. Since clients can make requests simultaneously, if there is no coordination in the "client-to-server" assignment process some clients may have to wait longer if they are assigned simultaneously to a single server when overall performance could improve had they been assigned to separate available servers. Therefore a principal approach to optimizing throughput when assigning client requests to supporting servers, is to keep the client queue sizes well balanced so as to prevent servers from having to remain idle while other servers are being overused. There are several potential examples of such systems involving data gathering and forwarding among sensors in a sensor network or when the servers are base-stations and the clients may be either rotating satellites or other wireless devices, for example. In this paper we consider the problem of finding an assignment of clients to servers that results in all clients receiving a packet while optimally balancing the sizes of remaining queues at the clients. We give a polynomial time algorithm for solving this problem which requires O((m + n)³n) arithmetic operations, where m is the number of client queues and n is the number of servers.

We consider the problem of k-partitioning a graph into k balanced subsets that minimizes the number of crossing weighted edges Previously, Maximum flow approach, iterative improvement, geometric representation were given to solve this problem. In this paper we propose a balanced k-partition heuristic based on a max spanning tree method.