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

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Summary

Introduction and Basic Definitions

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

VERKHOVSKY
Separation of “Even” and “Odd” Coefficients in AHP
Reduction of Algebraic Additions
Comparison of Evaluated Polynomials in TCA vs AHP
Toom-Cook Algorithm Compute
10. Multistage Implementation of TCA and AHP
10.2. Multi-Stage Implementation
12. Analysis of TCA vs AHP
13. Generalized Horner Rule for Homogeneous Polynomial
15. Optimized AHP
16. Conclusion
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