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The Accelerated Euclidean Algorithm

The Accelerated Euclidean Algorithm. Sidi Mohamed Sedjelmaci Proceedings of the EACA, (2004) 283-287 EACA : Encuentro de Algebra Computational y Aplicaciones. Outline. Algorithm ILE Algorithm Example Algorithm AEA Algorithm Example Complexity. Algorithm ILE.

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The Accelerated Euclidean Algorithm

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  1. The Accelerated Euclidean Algorithm Sidi Mohamed Sedjelmaci Proceedings of the EACA, (2004) 283-287 EACA : Encuentro de Algebra Computational y Aplicaciones

  2. Outline • Algorithm ILE • Algorithm • Example • Algorithm AEA • Algorithm • Example • Complexity

  3. Algorithm ILE • The algorithm ILE runs the extended Euclidean algorithm and stops when the remainder has roughly the half size of the inputs. • S.M., Sedjelmaci. “On A Parallel Lehmer-Euclid GCD Algorithm,” in Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’2001), 2001,303-308.

  4. Algorithm ILE

  5. Algorithm ILE • Example : U = 956722 V = 591286 n = l(U) = 20 p = l(V) = 20 m = p - - 1 = 20 – 10 – 1 = 9 (29 = 512)

  6. Algorithm AEA

  7. Algorithm AEA Step1. First, consider the 2r leading bits and reduce them to their half and obtain a matrix N1 . Strp2. Update the 2r next leading bits by N1 . Step3. Consider the new 2r leading bits and reduce them to their half and obtain a matrix N2 . Step4. Update the new r next leading bits by N2 . Step5. Compute M = N1* N2, this matrix is used to reduce the next 4r leading to their half.

  8. Algorithm AEA

  9. Algorithm AEA • Example : U = 956,722,026,041 V = 591,286,729,879 n = l(U) = 40 p = l(V) = 40 m = p - - 1 = 40 – 20 – 1 = 19 (219 = 524288)

  10. Algorithm AEA

  11. Algorithm AEA Step1,2 Step3,4

  12. Algorithm AEA U = 1134903 V = 701408 n = l(U) = 21 p = l(V) = 20 m = p - - 1 = 20 – 11 – 1 = 8 (28 = 256)

  13. Algorithm AEA Step5 U’ = 2178309 V’ = 1346269

  14. Algorithm AEA Let t(r) denoted the time cost for reducing 2r-bit inputes to their half. • Step1: t(r) • Strp2: O(M(r)) • Step3: t(r) • Step4: O(M(r)) • Step5: O(M(r)) • Step1-5:2t(r)+O(M(r)) • The total time complexity t(n)=O(logn M(n))

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