The posterior error estimation of an approximate solution for the integral equation φ(x)=f(x)+λ∫k(x,s)φ(s)ds is well known to be given by the formula max|e(x)|≤max|g(x)|/(1-M)for M≡|λ|max∫|k(x,s)|ds<1. Posterior error estimation was carried out by estimating the upper bound ε=max|h(x)|/(1-ν) for ν<1 regardless of the size of M. The overestimation in ε was evaluated by 2ν/(1-ν)relatively and by 2νε/(1+ν) absolutely. If we obtain a smaller parameter ν, the overestimation can be improved. Here, f(x), λ, and k(x,s) are given functions or parameters, g(x), h(x), and ν are calculable from known functions and the resolvent of the approximate kernel. In order to satisfy the condition ν<1, we easily evaluated the resolvent kernel using Bateman's theory. We evaluated this method by using some examples.

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For the universally composable (UC) security, verification methods based on Dolev-Yao symbolic model were proposed. In this paper, we demonstrate that our symbolic model can verify the UC security. More preciously, we define symbolic criteria in our model, corresponding to universally composable mutual authentication and key exchange protocols that use public key encryption and digital signature.

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Given a graph G=(V,E) with two designated vertices s, t∈V, assign integer 1 to s and n=|V| to t, and {2,3,…,n-1} to V-{s,t} so that each vertex in V-{s,t} has at least one smaller neighbor and one larger neighbor. Such assignment is called an st-numbering. It is known that if G is biconnected then an st-numbering always exists, and an algorithm to find an st-numbering is known. In this paper we give a simple but efficient algorithm to enumerate all st-numberings of a given biconnected plane graph G with designated two vertices s and t on the outer face.

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When Z/PZ (P is an odd prime) is decomposed into suitable two sets, Chebyshev polynomials have properties that each set is invariant under their operations and that for their degrees, they show different periods corresponding to each set. We construct a public key cryptography of RSA type based on Chebyshev polynomial by using the properties. Compared with RSA cryptography, our cryptography has possibility of realizing higher security and its deciphering time increases merely about 10%. However compared with fast RSA cryptography, it is about 4.5 times.

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This paper proposes a method to analyze asymmetric relational data in the framework of classical multidimensional scaling (MDS). Concretely, it represents each object by a pair of points and reformulates the classical MDS as an optimization problem. In particular, this formulation reduces to the traditional classical MDS if asymmetry is lacking. Further, actual numerical results are shown and the interpretation of asymmetry is given from the plotting results.

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