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An important problem that arises in many engineering applications is the boundary value problem for ordinary differential equations. There have been many computational methods proposed for dealing with this problem. The convergence of the iterative schemes to a true solution, when one such exists, and their numerical stability are the central issues discussed in the literature. In this paper, we discuss a method for approximating the vector field, maintaining the boundary conditions and numerical stability. If a true solution exists, a subsequence of solutions convergent to one such can be produced, by finer discretization of the solution space.

An integer adder for integers in the binary representation is one of the basic operations of any digital processor. For adding two integers of N bits each, the serial adder takes as many clock ticks. For achieving higher speeds, parallel circuits are discussed in the literature, and these circuits usually operate in two levels. At the lower level, integers represented by blocks of smaller number of bits are added, and in a cascade of stages in the next level, the carries produced in previous addition operations are summed to the augends. In this paper, we describe a fast method and an improvement of it. The first attempt resembles the operation method of the merge sort algorithm, from which some important properties of carries produced in each stage are analysed and assimilated, resulting in a parallel adder that runs in time comparable to the existing methods. After that, the crucial insights are brought to fruition in an improved design, which takes 2 clock ticks to perform the addition operation requiring only O(square(N)) space. The number of bits N is chosen usually to be a positive integer power of 2. The speedup is achieved by special purpose circuits for increment operations by i-th power of 2 , for i = 0, 1, ..., N-1, each operation taking only a single clock tick to complete. The usefulness of this adder for multiplication operation is discussed. The standard multiplication method utilizes quantizer and 3-bit to 2-bit consolidation circuits to produce an integer that represents in binary the number of 1s in a column corresponding to a place (weighted coefficient) of nonnegative integer power of 2. The last two consolidated integers are added by an adder in the end.

In this paper, algorithms for multivariate public key cryptography and digital signature are described. Plain messages and encrypted messages are arrays, consisting of elements from a fixed finite ring or field. The encryption and decryption algorithms are based on multivariate mappings. The security of the private key depends on the difficulty of solving a system of parametric simultaneous multivariate equations involving polynomial or exponential mappings. The method is a general purpose utility for most data encryption, digital certificate or digital signature applications. For security protocols of the application layer level in the OSI model, the methods described in this paper are useful., Comment: arXiv admin note: substantial text overlap with arXiv:1608.06472

In this paper, algorithms for multivariate public key cryptography and digital signature are described. Plain messages and encrypted messages are arrays, consisting of elements from a fixed finite ring or field. The encryption and decryption algorithms are based on multivariate mappings. The security of the private key depends on the difficulty of solving a system of parametric simultaneous multivariate equations involving polynomial or exponential mappings. The method is a general purpose utility for most data encryption, digital certificate or digital signature applications.