This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
The Montgomery Modular Inverse-Revisited
July 2000 (vol. 49 no. 7)
pp. 763-766

Abstract—We modify an algorithm given by Kaliski to compute the Montgomery inverse of an integer modulo a prime number. We also give a new definition of the Montgomery inverse, and introduce efficient algorithms for computing the classical modular inverse, the Kaliski-Montgomery inverse, and the new Montgomery inverse. The proposed algorithms are suitable for software implementations on general-purpose microprocessors.

[1] W. Diffie and M.E. Hellman, New Directions in Cryptography IEEE Trans. Information Theory, vol. 22, pp. 644-654, 1976.
[2] Ö. Egecioglu and Ç. K. Koç, "Exponentiation Using Canonical Recoding," Theoretical Computer Science, vol. 129, no. 2, pp. 407-417, 1994.
[3] B.S. Kaliski Jr., “The Montgomery Inverse and Its Applications,” IEEE Trans. Computers, vol. 44, no. 8, pp. 1,064-1,065, Aug. 1995.
[4] D. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1998.
[5] N. Koblitz, “Elliptic Curve Cryptosystems,” Math. of Computation, vol. 48, no. 177, pp. 203-209, Jan. 1987.
[6] A.J. Menezes, Elliptic Curve Public Key Cryptosystems. Boston: Kluwer Academic, 1993.
[7] P.L. Montgomery, “Modular Multiplication without Trial Division,” Math. of Computation, vol. 44, no. 170, pp. 519-521, Apr. 1985.
[8] Nat'l Inst. for Standards and Tech nology, Digital Signature Standard (DSS). Federal Register, 56:169, Aug. 1991.
[9] J.-J. Quisquater and C. Couvreur, “Fast Decipherment Algorithm for RSA Public-Key Cryptosystem,” Electronics Letters, vol. 18, no. 21, pp. 905-907, Oct. 1982.
[10] M. Rosing, Implementing Elliptic Curve Cryptography. Manning Publications, 1999.
[11] R. Schroeppel, S. O'Malley, H. Orman, and O. Spatscheck, “A Fast Software Implementation for Arithmetic Operations in GF($2^n$),” Proc. Advances in Cryptology–CRYPTO '95, pp. 43-56, 1995.

Index Terms:
Modular arithmetic, modular inverse, almost inverse, Montgomery multiplication, cryptography.
Citation:
E. Savas, Ç.k. Koç, "The Montgomery Modular Inverse-Revisited," IEEE Transactions on Computers, vol. 49, no. 7, pp. 763-766, July 2000, doi:10.1109/12.863048
Usage of this product signifies your acceptance of the Terms of Use.