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Multiplication of large integers by the use of modular arithmetic: application to cryptography

Multiplication of large integers by the use of modular arithmetic: application to cryptography Computing the long multiplication in fixed-radix representation is described first which suggests the use of two mixed solutions: first the sequentialisation of Karatsuba's algorithm by its extension to hexa and octo-mul then their judicious combination plus Implementation in Occam 2 language. Computing the long multiplication in modular representation . Including the principles of modular arithmetic and the Chinese remainder theorem, with efficient methods, is given in detail, together with their implementation for transformation from integer fixed-radix to modular and back again . Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm. Montgomery's method for "Computing the long multiplication without trial division" for avoiding time consuming integer-modular-integer conversions at each multiplication-square step is described. It is possible to apply such a method to the exponentiation process currently used in Cryptography because It is interesting to use it only in the case where several multiplications are done in modulo N. Cryptographic applications for computing asymmetric keys may use the present implementations to advantage. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM SIGSAC Review Association for Computing Machinery

Multiplication of large integers by the use of modular arithmetic: application to cryptography

ACM SIGSAC Review , Volume 7 (4) – Jan 1, 1990

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Publisher
Association for Computing Machinery
Copyright
Copyright © 1990 by ACM Inc.
ISSN
0277-920X
DOI
10.1145/382089.382682
Publisher site
See Article on Publisher Site

Abstract

Computing the long multiplication in fixed-radix representation is described first which suggests the use of two mixed solutions: first the sequentialisation of Karatsuba's algorithm by its extension to hexa and octo-mul then their judicious combination plus Implementation in Occam 2 language. Computing the long multiplication in modular representation . Including the principles of modular arithmetic and the Chinese remainder theorem, with efficient methods, is given in detail, together with their implementation for transformation from integer fixed-radix to modular and back again . Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm. Montgomery's method for "Computing the long multiplication without trial division" for avoiding time consuming integer-modular-integer conversions at each multiplication-square step is described. It is possible to apply such a method to the exponentiation process currently used in Cryptography because It is interesting to use it only in the case where several multiplications are done in modulo N. Cryptographic applications for computing asymmetric keys may use the present implementations to advantage.

Journal

ACM SIGSAC ReviewAssociation for Computing Machinery

Published: Jan 1, 1990

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