The Montgomery Inverse and Its Applications

@article{Kaliski1995TheMI,
  title={The Montgomery Inverse and Its Applications},
  author={Burton S. Kaliski},
  journal={IEEE Trans. Computers},
  year={1995},
  volume={44},
  pages={1064-1065}
}
  • B. Kaliski
  • Published 1 August 1995
  • Mathematics
  • IEEE Trans. Computers
The Montgomery inverse of b module a is b/sup -1/2/sup n/ mod a, where n is the number of bits in a. The right-shifting binary algorithm for modular inversion is shown naturally to compute the new inverse in fewer operations than the ordinary modular inverse. The new inverse facilitates recent work by Koc on modular exponentiation and has other applications in cryptography. > 
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References

SHOWING 1-10 OF 13 REFERENCES

High-radix and bit recoding techniques for modular exponentiation

  • Ç. Koç
  • Computer Science
    Int. J. Comput. Math.
  • 1991
It is shown that the high-radix methods with optimal choice of the radix provide significant reductions in the number of multiplications required for modular exponentiation, and bit recoding techniques similar to those used in binary multiplication algorithms can be used to further reduce the total number ofmultiplications.

Modular multiplication without trial division

A method for multiplying two integers modulo N while avoiding division by N, a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms.

A redundant binary Euclidean GCD algorithm

An efficient implementation of the Euclidean GCD (greatest common divisor) algorithm employing the redundant binary number system is described, which is competitive with floating point division in terms of the number of shift and add/subtract operations.

Optimized implementation of RSA cryptosystem

Fast implementations of RSA cryptography

The authors detail and analyze the critical techniques that may be combined in the design of fast hardware for RSA cryptography: chinese remainders, star chains, Hensel's odd division, carry-save representation, quotient pipelining, and asynchronous carry completion adders.

A Cryptographic Library for the Motorola DSP56000

A cryptographic library for the Motorola DSP56000 that provides hardware speed yet software flexibility, and an algorithm for modular multiplication that interleaves multiplication with Montgomery modular reduction to give a very fast implementation of RSA.

Fast decipherment algorithm for RSA public-key cryptosystem

A fast algorithm is presented for deciphering cryptograms involved in the public-key cryptosystem proposed by Rivest, Shamir and Adleman, based on the Chinese remainder theorem and on improved modular multiplication algorithms.

A public key cryptosystem and a signature scheme based on discrete logarithms

The invention pertains to a novel method of forming a flexible, tapered end on a thermoplastic molding strip that is flexible and virtually free of residual stress which may cause a strip to delaminate when applied to a desired substratum.

Systolic Modular Multiplication

A simple systolic array for achieving the effect of modular reduction, in linear time, is described. This circuit, in conjunction with Atrubin’s multiplier, performs modular multiplication in linear…

The Art of Computer Programming

The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.