Division (mathematics)

About: Division (mathematics) is a research topic. Over the lifetime, 12717 publications have been published within this topic receiving 87814 citations.

Fast recursive division

Abstract: We present a new recursive method for division with remainder of integers. Its running time is 2K(n)+O(n log n) for division of a 2n-digit number by an n-digit number where K(n) is the Karatsuba multiplication time. It pays in practice for numbers with 860 bits or more. Then we show how we can lower this bound to 3=2K(n)+O(n log n) if we are not interested in the remainder. As an application of division with remainder we show how to speedup modular multiplication. We also give practical results of an implementation that allow us to say that we have the fastest integer division on a SPARC architecture compared to all other integer packages we know of.

Wavelength division multi/demultiplexer

Abstract: A wavelength division multi/demultiplexer multi/demultiplexes incident light to emit demultiplexed lights from each of a plurality of light output ends and includes a first wavelength division multi/demultiplexing circuit (19) having at least one light input end (2) and a plurality of light output end (6); and second wavelength division multi/demultiplexing circuits (20) of AWG type formed on a substrate, including a plurality of light input and output ends (2,6), one or two light multiplexing and demultiplexing sections (3,5), and a plurality of arrayed waveguides (4), and having a transmission bandwidth different from that of the first circuit, the transmission bandwidth indicating the width of a light frequency that can pass through each light input and output end, wherein the first circuit and at least one of the second circuits are optically connected together (21), and wherein the light output ends of the final second circuit are used as the light output ends of the wavelength division multi/demultiplexer. It is possible to greatly increase the number of channels of the wavelength division multi/demultiplexer using the AWG type wavelength division multi/demultiplexing circuit.

MILP-aided Method of Searching Division Property Using Three Subsets and Applications

Abstract: Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and then conventional bit-based division property (CBDP) and bit-based division property using three subsets (BDPT) were proposed by Todo and Morii at FSE 2016. At the very beginning, the two kinds of bit-based division properties once couldn’t be applied to ciphers with large block size just because of the huge time and memory complexity. At ASIACRYPT 2016, Xiang et al. extended Mixed Integer Linear Programming (MILP) method to search integral distinguishers based on CBDP. BDPT can find more accurate integral distinguishers than CBDP, but it couldn’t be modeled efficiently.

Very high radix square root with prescaling and rounding and a combined division/square root unit

Abstract: An algorithm for square root with prescaling and selection by rounding is developed and combined with a similar scheme for division. Since division is usually more frequent than square root, the main concern of the combined implementation is to maintain the low execution time of division, while accepting a somewhat larger execution time for square root. The algorithm is presented in detail, including the mathematical development of bounds for the first square-root digit and for the scaling factor. The proposed implementation is described, evaluated and compared with other combined div/sqrt units. The comparisons show that the proposed scheme potentially produces a significant speed-up for division, whereas, for square root, the speed-up is small.

Model Order Reduction by Using the Balanced Truncation and Factor Division Methods

Abstract: The aim of this paper is to construct a new model order reduction method for linear dynamic systems. In this technique, the denominator polynomial of the reduced order model (ROM) is obtain...