Mausumi Bose

Optimal (k, n) visual cryptographic schemes for general k

In (k, n) visual cryptographic schemes (VCS), a secret image is encrypted into n pages of cipher text, each printed on a transparency sheet, which are distributed among n participants. The image can be visually decoded if any k(≥2) of these sheets are stacked on top of one another, while this is not possible by stacking any k − 1 or fewer sheets. We employ a Kronecker algebra to obtain necessary and sufficient conditions for the existence of a (k, n) VCS with a prior specification of relative contrasts that quantify the clarity of the recovered image. The connection of these conditions with an L1-norm formulation as well as a convenient linear programming formulation is explored. These are employed to settle certain conjectures on contrast optimal VCS for the cases k = 4 and 5. Furthermore, for k = 3, we show how block designs can be used to construct VCS which achieve optimality with respect to the average and minimum relative contrasts but require much smaller pixel expansions than the existing ones.

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k+ 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.

Optimal (2, n) visual cryptographic schemes

In (2,n) visual cryptographic schemes, a secret image(text or picture) is encrypted into n shares, which are distributed among n participants. The image cannot be decoded from any single share but any two participants can together decode it visually, without using any complex decoding mechanism. In this paper, we introduce three meaningful optimality criteria for evaluating different schemes and show that some classes of combinatorial designs, such as BIB designs, PBIB designs and regular graph designs, can yield a large number of black and white (2,n) schemes that are optimal with respect to all these criteria. For a practically useful range of n, we also obtain optimal schemes with the smallest possible pixel expansion.

Key predistribution schemes for distributed sensor networks via block designs

Key predistribution schemes for distributed sensor networks have received significant attention in the recent literature. In this paper we propose a new construction method for these schemes based on combinations of duals of standard block designs. Our method is a broad spectrum one which works for any intersection threshold. By varying the initial designs, we can generate various schemes and this makes the method quite flexible. We also obtain explicit algebraic expressions for the metrics for local connectivity and resiliency. These schemes are quite efficient with regard to connectivity and resiliency and at the same time they allow a straightforward shared-key discovery.

Replacement sort revisited: The “gold standard” unearthed!

Abstract The present paper shows that for certain algorithms such as sorting, the parameters of the input distribution must also be taken into account, apart from the input size, for a more precise evaluation of computational and time complexity (average case only) of the algorithm in question (the so-called “gold standard”). Some concrete results are presented to warrant a new and improved model for replacement sort (also called selection sort) as T avg ( n , p 1 , p 2 , … p k ) = a 0 + b 0 n ( n - 1 ) / 2 + c 0 i ( n , p 1 , p 2 , … p k ) + ϵ , where the LHS gives the average case time complexity, n is the input size, p i ’s the parameters of the input distribution characterizing the sorting elements, i is the average number of interchanges which is a function of both the input size and the parameters, the rest of the terms arising due to linear regression and have usual meanings. The error term ϵ arises as we have fixed only the input size n in the model but varying the specific input elements and their relative positions in the array, for a particular distribution [H. Mahmoud, Sorting: A Distribution Theory, John Wiley and Sons, 2000]. The term n ( n - 1 ) / 2 represents the number of comparisons. We claim this to be an improvement over the conventional model, namely, T avg ( n ) = a + bn + cn 2 + ϵ , which stems from the O ( n 2 ) complexity for this algorithm. We argue that the new model in our opinion can be a guiding factor in distinguishing this algorithm from other sorting algorithms of similar order of average complexity such as bubble sort and insertion sort. Note carefully that the dependence of the number of interchanges on the parameters is more prominent for discrete distributions rather than continuous ones and we suspect this to be because the probability of a tie is zero in a continuous case. However, presence of ties and their relative positions in the array is crucial for discrete cases. And this is precisely where the parameters of the input distribution come into play. Those algorithms where ties have a greater influence on some of the computations will have greater influence of parameters of the input distribution in it. Another strength of the paper is that it brings up the close connection between algorithmic complexity and computer experiments , a crucial issue which is overlooked in the textbooks on algorithms . This is a paper on modeling rather than speed.

Optimal crossover designs under a general model

Some assumptions are implicit in the traditional model used for studying the optimality properties of cross-over designs. Many of these assumptions might not be satisfied in experimental situations where these designs are to be applied. In this paper, we modify the model by relaxing these assumptions and show a class of designs to be universally optimal under the modified model.

Universal optimality of Patterson’s crossover designs

We show that the balanced crossover designs given by Patterson [Biometrika 39 (1952) 32-48] are (a) universally optimal (UO) for the joint estimation of direct and residual effects when the competing class is the class of connected binary designs and (b) UO for the estimation of direct (residual) effects when the competing class of designs is the class of connected designs (which includes the connected binary designs) in which no treatment is given to the same subject in consecutive periods. In both results, the formulation of UO is as given by Shah and Sinha [Unpublished manuscript (2002)]. Further, we introduce a functional of practical interest, involving both direct and residual effects, and establish (c) optimality of Pattersons designs with respect to this functional when the class of competing designs is as in (b) above.

MULTINOMIAL SUBSET SELECTION USING INVERSE SAMPLING AND ITS EFFICIENCY WITH RESPECT TO FIXED SAMPLING

The multinomial selection problem is considered in its general form where the objective is to select a subset of s cells which contain the t ‘best’ cells, s ≥ t. The inverse-sampling procedure is studied for this problem and the LFC is derived under the difference zone. An expression for the relative efficiency of this procedure with respect to the widely used fixed-sample-size selection procedure is obtained and theoretical bounds are derived for this efficiency. It is found that the inverse-sampling procedure performs uniformly better than the usual fixed-sampling procedure in the case s = t and is often more efficient for s > t. When the selection goal is to select any c of the t best cells, using a subset of s cells, expressions for efficiency may be similarly obtained.

Designs with Repeated Measurements on Any Number of Units over Varying Periods

In the usual repeated measurements designs (RMDs), the subjects are all observed for the same number of periods and the optimum RMDs require specified numbers of subjects, usually depending on the number of treatments to be used. In practice, it is sometimes not feasible to meet these requirements. To overcome this problem, alternative designs are suggested where any number of available subjects may be used and they may be observed for different periods. These designs are based on suitable serially balanced sequences which are shown to be optimal. Moreover, besides the usual direct and residual effects, the model considered has an extra term due to the interaction effect between them. The recommended designs are universally optimal in a very general class.

Optimal main effect plans in blocks and related nested row-column designs

Abstract Optimal main effect plans with nonorthogonal blocks of size two each are constructed. Using these block designs, optimal block designs with nested rows and columns are constructed for some main effect plans.