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# The Main Features of Modular Arithmetic and Its Importance

Modular arithmetic can be used to computeexactly at low cost a set of simple computationsThese include most geometric predicates thatneed to be checked exactly and especially thesign of determinants and more general polynomialModular arithmetic resides on the ChineseRemainder Theorem which states that whencomputing an integer expression you only have tocompute it modulo several relatively prime integerscalled the modulis The true integer value can thenbe deduced but also only its sign in a simple andThe main drawback with modular arithmetic is itsstatic nature because we need to have a bound onthe result to be sure that we preserve ourselvesfrom overflows that cant be detected easilywhile computing The smaller this known bound isthe less computations we have to do We have developped a set of efficient tools to dealwith these problems and we propose a filteredapproach that is an approximate computationusing floating point arithmetic followed in the badcase by a modular computation of the expressionof which we know a bound thanks to the floatingpoint computation we have just done Theoreticalwork has been done in common with Victor Panand See the bibliography for details At the moment only the tools to compute withoutfilters are available The aim is now to build acompiler that produces exact geometricpredicates with the following scheme filter modular computation This approach is notcompulsory optimal in all cases but it has theadvantage of simpleness in most geometric testsConcerning the implementation the ModularPackage contains routines to compute sign ofdeterminants and polynomial expressions usingmodular arithmetic It is already usable tocompute signs of determinants in any dimensionwith integer entries of less than 53 bits In thenear future we plan to add a floating point filterBibliographyExplains basically the definition of modular arithmetic and contents of it

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