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Fast Multicomputation with Asynchronous Strategy

Fast Multicomputation with Asynchronous Strategy. IEEE Trans. On Computers. Vol. 56 NO. 2, FEB. 2007 Wu-C. Yang, D.J. Guan, and C.S. Laih 報告人 : 9603007D 洪清波 報告大綱 : 一 . Introduction 二 . Recoding Method 三 . Asynchronous Strategy for Sparse Forms SS 1 , DS 1.

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Fast Multicomputation with Asynchronous Strategy

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  1. Fast Multicomputation with Asynchronous Strategy IEEE Trans. On Computers. Vol. 56 NO. 2, FEB. 2007 Wu-C. Yang, D.J. Guan, and C.S. Laih 報告人: 9603007D 洪清波 報告大綱: 一. Introduction 二. Recoding Method 三. Asynchronous Strategy for Sparse Forms SS1, DS1

  2. 一. Introduction(1/3) 公開金鑰: ECC: xA + yB, RSA: axby Exam: 8410 = (01010100)2 = x xi compute result W(xi) 0 c = 2 * 0 0 0 1 c = 2c + A A 1 0 c = 2c 2A 0 1 c = 2c + A 5A 1 0 c = 2c 10A 0 1 c = 2c + A 21A 1 0 c = 2c 42A 0 0 c = 2c 84A 0

  3. 一. Introduction(2/3) 計算: xA + yB Exam: x = (01010)2 = 1010 y= (01110)2 = 1410 xi yi compute result W(xi,yi) 0 0 c = 2 * 0 0 0 1 1 c = 2c + A + B A+B 1 0 1 c = 2c + B 2A+3B 1 1 1 c = 2c + A + B 5A+7B 1 0 0 c = 2c 10A+14B 0 Synchronous: (xi,yi), i = j Asynchronous: (xi,yi), i ≠ j

  4. 一. Introduction(3/3) Improvement Strategy: Pre-computing (memory sufficient) Recoding: Two Sparse Form, DJM, JSF (memory limited) Asynchronous: SS1, DS1 (memory limited)

  5. 二. Recoding Method(1/3) 1. B. S. D (binary signed-digit) expected W(x) = 1/3n Two Sparse Form expected W(x,y)=(1 – (2/3)2)n = 5/9n ≈ 0.556n

  6. 二. Recoding Method(2/3) 2. D. J. M method D. J. M expected W(x,y) ≈ 0.518n

  7. 二. Recoding Method(3/3) 3. J. S. F method (Joint Sparse Form) J. S. F expected W(x,y) ≈ 0.5n when n→∞

  8. 1. SS1 method 三. Asynchronous Strategy for Sparse Forms

  9. 1. SS1 method example: 計算: xA + yB Exam: x = (01010100)2 = 8410 y= (0010- 0- 0)2 = 2210 i 7 6 5 4 3 2 1 0 state SS1 S0 S0→Sx Sx Sx Sx Sx Sx Sx→S0 compute SS1 c=2*0 2c+A 2c 2c+A+2B 2c 2c+A-2B 2c 2(c-B) BSD c=2*0 2c+A 2c+B 2c+A 2c-B 2c+A 2c-B 2c result SS1 0 A 2A 5A+2B 10A+4B 21A+6B 42A+12B 84A+22B BSD 0 A 2A+B 5A+2B 10A+3B 21A+6B 42A+11B 84A+22B W(x,y) SS1 0 1 0 1 0 1 1 0 BSD 0 1 1 1 1 1 1 0 三. Asynchronous Strategy for Sparse Forms

  10. State S0: in line 5 W1 = W2; in line 6 if xi = 1 then W2 decreased by 1 State Sx: xi:0 yj:î 0 î 0 î0 î0 î 0 î0 î0 î0 î0 î 0 …0 î… î k (k-1)/2 k:1, 3, 5, … xi:î yj: 0 î 0 î0 î 0 î0 î0 î 0 î 0 î0 î0 î… î 0 …0 k (k+1)/2 k:1, 3, 5, … xi:0 yj:0 0 î î 0 0 î0 î î0 î 0 0 î0 î0 î î0 î 0 î 0 0 …î î… 0 k  k/2 k:2, 4, 6, … xi:î yj:î î 0 0 î î0 î 0 0 î0 î î0 î 0 î 0 0 î0 î0 î î… 0 0 …î k  k/2 k:2, 4, 6, …

  11. Pa|b a: ith digit, b: (i + 1)st xi+1xixi-1 Parse form: P0 = 2/3, PÎ = 1/3 Proof: no two consecutive digits nonzero. a. Let P0|0 = p, PÎ|0 = 1 – p b. P0 = P0 • P0|0 + PÎ • P0|Î  2/3 = 2/3• p + 1/3• 1  p = 1/2

  12. Paα|bβ where xi = a, xi+1 = b, yi = α, yi+1 = β Proof: x and y are independent  Paα|bβ = Pa|b• Pα|β

  13. Pbβ Paα|bβ 4/9 1/4 “ “ “ “ “ “ 1/9 1 2/9 ½ “ “ “ “ “ “

  14. DS1 method (1/3)

  15. DS1 method (2/3)

  16. DS1 method (3/3)

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