Abstract
Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper. The homogeneity of polynomials provides several simplifications: reduction of system of equations and elimination of necessity to evaluate polynomials in points with larger coordinates. It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation. A multistage implementation of these algorithms is also less efficient than their direct implementation. Although the proposed algorithms as well as the corresponding Toom-Cook algorithms require numerous algebraic additions, the Generalized Horner rule for evaluation of homogeneous polynomials, provided in the paper, decrease this number twice.
Highlights
Introduction and Basic DefinitionsCrypto-immunity of various protocols of secure communication over open channels is based on modular arithmetic of large integers with hundreds of decimal digits
Several algorithms based on homogeneous polynomials for multiplication of large integers are described in the paper
It is demonstrated that a two-stage implementation of the proposed and Toom-Cook algorithms asymptotically require twice as many standard multiplications than their direct implementation
Summary
Crypto-immunity of various protocols of secure communication over open channels is based on modular arithmetic of large integers with hundreds of decimal digits. Standard programming libraries in general-purpose computers handle multiplication of integers A and B if the number of decimal digits in each does not exceed m. Such integers we will refer to as standard integers. Analysis of computational complexity of Toom-Cook algorithm (TCA) is provided in [5] and theoretical foundation for efficient multiplication of large integers is discussed in [6]. A special case of the TCA, where one multiplier is significantly larger than another, is considered in [9]. B = 608,348,696,284; using a computing device that cannot multiply integers of order higher than O 103
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