New Modified Euclidean and Binary Greatest Common Divisor Algorithm

REFERENCES

• R. P. Brent and P. Zimmermann, Modern Computer Arithmetic, Cambridge Monographs on Computational and Applied Mathematics. New York: Cambridge University Press, 2010.
   Google Scholar
• A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography. Boca Raton, FL: CRC Press, 1997.
   Google Scholar
• H. Cohen, G. Frey, R. Avanzi, T. Lange, and N. Kim, Handbook of Elliptic and Hyperelliptic Curve Cryptography, 2nd ed. New York: Chapman and Hall/CRC, 2005.
   Google Scholar
• B. Vallée, “The complete analysis of the binary Euclidean algorithm,” Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423, pp. 77–94, Jun. 1998.
   Google Scholar
• J. P. Sorenson, “Two fast GCD algorithms,” J. Algorithms, Vol. 16, pp. 110–44, Jan. 1994.
   Web of Science ®Google Scholar
• J. P. Sorenson, “An analysis of Lehmer's Euclidean GCD algorithm,” in ACM International Symposium on Symbolic and Algebraic Computation, Montreal, Canada, 1995, pp. 254–8.
   Google Scholar
• D. H. Lehmer, “Euclid's algorithm for large numbers,” Am. Math. Monthly, Vol. 45, pp. 227–33, Apr. 1938.
   Google Scholar
• S. N. Parikh and D. W. Matula, “A redundant binary euclidean GCD algorithm,” in Proc. 10th IEEE Symposium on Computer Arithmetic, Grenoble, France, 1997, pp. 220–5.
   Google Scholar
• T. Jebelean, “Comparing several GCD algorithms,” in Proc. 11th Symposium on Computer Arithmetic, Windsor, Ontario, Canada, 1993, pp. 180–5.
   Google Scholar
• T. Jebelean, “A generalization of the binary GCD algorithm,” in Proc. Int. Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–6.
   Google Scholar
• K. Weber, “The accelerated integer GCD algorithm,” ACM Trans. Math. Software, Vol. 21, no. 1, pp. 111–22, Mar. 1995.
   Web of Science ®Google Scholar
• D. E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, Vol. 2, 3rd ed. Reading, MA: Addison-Wesley, 1998.
   Google Scholar
• K. Weber, “Parallel implementation of the accelerated integer GCD algorithm,” J. Symbol. Comput. Special Issue Parallel Symbol. Comput., Vol. 21, no. 4–6, pp. 457–66, Apr. 1996.
   Web of Science ®Google Scholar
• S. Damien and P. Zimmermann, “A binary recursive GCD algorithm,” Lecture Notes Comput. Sci. Algorithm. Number Theory, Vol. 3076, pp. 411–25, Jun. 2004.
   Web of Science ®Google Scholar
• C. K. Yap, Fundamental Problem in Algorithmic Algebra. USA: Oxford University Press, 2000.
   Google Scholar
• S. M. Sedjelmaci, “A modular reduction for GCD computation,” J. Comput. Appl. Math., Vol. 162, pp. 17–31, Jan. 2004.
   Web of Science ®Google Scholar
• J. Stein, “Computational problem associated with Racah algebra,” J. Comput. Phys., Vol. 1, pp. 397–405, Feb. 1967.
   Google Scholar
• S. M. Sedjelmaci, “The mixed binary Euclidean algorithm,” Electron. Notes Discrete Math., Vol. 35, pp. 169–79, Dec. 2009.
   Google Scholar
• J. P. Sorenson, “An analysis of the generalized binary GCD algorithm,” in High Primes and Misdemeanors: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams. Providence, RI: American Mathematical Society, 2004, pp. 327–40.
   Google Scholar
• G. Cesari, “Parallel implementation of Schönhage's integer GCD algorithm,” ANTS-III, LNCS, Vol. 1423, pp. 64–76, Jun. 1998.
   Google Scholar
• B. Chor and O. Goldreich, “An improved parallel algorithm for integer GCD,” Algorithmica, Vol. 5, pp. 1–10, Jun. 1990.
   Web of Science ®Google Scholar
• R. Kannan, G. Miller, and L. Rudolph, “Sublinear parallel algorithm for computing the greatest common divisor of two integers,” SIAM J. Comput., Vol. 16, pp. 7–16, Feb. 1987.
   Google Scholar
• S. M. Sedjelmaci, “A parallel extended GCD algorithm,” J. Discrete Algorithms, Vol. 6, pp. 526–38, Sep. 2008.
   Google Scholar
• K. A. Berman and J. L. Paul, “Algorithms: sequential, parallel, and distributed,” 1st ed. Richmond, TX: Thomson/Course Technology, 2004.
   Google Scholar
• T. Granlund, The GNU Multiple Precision Arithmetic Library: GNU Manual, 5.1.1 ed., 2013. Available: https://gmplib.org/gmp-man-5.1.1.pdf
   Google Scholar