1 a redundant binary euclidean gcd algorithm s. n. parikh, d. w. matula computer arithmetic 10th...

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A Redundant Binary Euclidean GCD Algorithm

S. N. Parikh, D. W. MatulaComputer Arithmetic10th IEEE Symposium

pp.220-2251991

陳正昇

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Outline Binary Euclidean Algorithm Redundant Binary Representation

and normalization Redundant Binary Euclidean

Algorithm

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Binary Euclidean Algorithm Normalized

數字介於 [1/2,1),(-1,-1/2] 之間 Time complexity

O(nlogn) with O(n) hardware

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Binary Euclidean Algorithm

1. While P and Q not normalized do //normalize P and Q 2. leftshift P,Q,Up,Uq; 3. While Q != 0 do begin //find GCD(P,Q) , terminates when Q=0 4. While Q not normalized do //normalize Q 5. leftshift Q and Uq; 6. Loop //find the remainder of P/Q 7. While P not normailzed do //normalize P 8. Exit Loop if Uq=Up; 9. Leftshift P and Up; 10. End; 11. If sign(P)=sign(Q) then //find the difference of P and Q 12. P=P-Q 13. else P=P+Q; 14. EndLoop; 15. Swap(P,Q);Swap(Up,Uq); //interchange of divisor and remain

der 16. End; 17. While Up != 1 do rightshift P and Up; //shift back the P

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1. While P and Q not normalized do //normalize P and Q 2. leftshift P,Q,Up,Uq; 3. While Q != 0 do begin //find GCD(P,Q) 4. While Q not normalized do //normalize Q 5. leftshift Q and Uq; 6. Loop //find the remainder of

P/Q 7. While P not normailzed do //normalize P 8. Exit Loop if Uq=Up; 9. Leftshift P and Up; 10. End;

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11. If sign(P)=sign(Q) then //find the difference of P and Q

12. P=P-Q 13. else P=P+Q; 14. EndLoop; 15. Swap(P,Q);Swap(Up,Uq); //interchange of divisor and rema

inder 16. End; 17. While Up != 1 do rightshift P and Up; //shift back the P

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EX. of Binary Euclidean Algorithm

P = 27(dec) = 11011(bin) 27/32 = 0.11011(bf)

Q = -12(dec) = -1100(bin) -12/32 = -0.011(bf)

GCD(P,Q) = 3(dec) = 11(bin) 3/32 = 0.00011(bf)

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Step( 演算法第幾行 )

P(dec) Q(dec) P(bin) Q(bin) Up Uq

27/32 -12/32 0.11011 -0.011 1 1

2. -24/32 -0.11 2

12. 3/32 0.00011

15. -24/32 3/32 -0.11 0.0001

1

2 1

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-24/32 3/32 -0.11 0.0001

1

2 1

5. 24/32 0.11 8

13. 0 0

15. 24/32 0 0.11 0 8 2

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3/32 = 0.00011(bf) GCD(P,Q) = 11(bin) = 3(dec)



24/32 0 0.11 0 8 2

17. 3/32 0.00011 1

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Redundant Binary Representation and normalization

Binary redundant representation

(b = 2, -1 aj 1 )

n = (1864)10 = (11101001000)2= (1001

01001000)2

1 = -1

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Signed bit fraction

b0.b1b3b4```bk , bi = 1,0,1 for all i

1 = -1

Normalized standard form :

b0=0 , b1!=0 and b2!=-b1

Complement standard form :

b0!=0 , b1=0

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Normalized 數字介於 (1/4,1),(-1,-1/4) 之間

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Simshift(P)=

0.b1b3b4```bk , if b1!=0 and b1=-b2

0.b2b3b4```bk , if b1=0

在 Redundant Binary 形式下的 leftshift (*2)

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decomp(P)=

0.b0b2b3```bk , if b0=-b1!=0

P , if b0=0

轉成 Normalized standard form

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a diff b= a-b for ab >= 0 (a,b 同號 ) a+b for ab < 0 (a,b 不同號 )

求 a 和 b 的差

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Redundant Binary Euclidean Algorithm

1. While P and Q not normalized do 2. Simshift P,Q and leftshift Up,Uq //Simshift P and Q 3. LoopA 4. While Q not normalized do //Simshift P 5. Simshift Q and leftshift Uq; 6. Exit LoopA if Uq overflows; 7. LoopB //find the remainder of P/Q 8. While P not normalized do //Simshift Q 9. Exit LoopB if Up=Uq; 10. Simshift P and leftshift Up; 11. If(P diff Q) is normalized then //find the difference of P and Q 12. If sign(P diff q) != sign(P) then 13. If Up != Uq then 14. leftshift Up; P = 2P diff Q; 15. Else exit LoopB; 16. Else P = P diff 2Q; 17. Else P = P diff Q; 18. P = decomp(P); //transform p to normalized standard for

m 19. Simshift P; //Simshift P 20. If Up != Uq then leftshift Up; 21. Else leftshift Up and exit LoopB; 22. End LoopB; 23. Swap(P,Q);Swap(Up,Uq); //interchange of divisor and remainder 24. End LoopA; 25. While Up != 1 do rightshift P,Up //shift back the P

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1. While P and Q not normalized do 2. Simshift P,Q and leftshift Up,Uq //Simshift P and Q 3. LoopA 4. While Q not normalized do //Simshift P 5. Simshift Q and leftshift Uq; 6. Exit LoopA if Uq overflows; 7. LoopB //find the remainder of P/Q 8. While P not normalized do //Simshift Q 9. Exit LoopB if Up=Uq; 10. Simshift P and leftshift Up;

//find the difference of P and Q

11. If(P diff Q) is normalized then 12. If sign(P diff q) != sign(P) then 13. If Up != Uq then 14. leftshift Up; P = 2P diff Q; 15. Else exit LoopB;

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16. Else P = P diff 2Q; 17. Else P = P diff Q;

//transform p to normalized standard form 18. P = decomp(P); 19. Simshift P; //Simshift P 20. If Up != Uq then leftshift Up; 21. Else leftshift Up and exit LoopB; 22. End LoopB;

//interchange of divisor and remainder 23. Swap(P,Q);Swap(Up,Uq); 24. End LoopA; 25. While Up != 1 do rightshift P,Up //shift back the P

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Time complexity Addition is O(n) 4-2 signed 1-bit

adders O(n) with O(n) bit level

processors

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EX. of Redundant Binary Euclidean Algorithm

P = 27(dec) = 011011(bin) 27/32 = 0.11011(sbf)

Q = -12(dec) = 111001(1`sc) -12/32 = 0.011(sbf) GCD(P,

Q) = 3(dec) = 000011(bin) 0.00011 (sbf) = 3/32

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P Q P(sbf) Q(sbf) Up Uq P diff

Q

27/32 -12/32 0.1101

1

0.011 1 1

11. 15/32

16. 3/32 0.0001

1

19. 6/32 0.0011

21. 2

23. -

12/32

6/32 0.011 0.0011 1 2

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Q

-

12/32

6/32 0.011 0.0011 1 2

5. 12/32 0.011 4

11. 0

17. 0 0.00

20. 2

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Q

0 12/32 0.00 0.011 2 4

10. 0 0.00 4 4

23. 12/32 0 0.011 0.00 4 4

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3/32 = 0.00011(bf) GCD(P,Q) = 011(bin) = 3(dec)



Q

12/32 0 0.011 0.00 4 4

5. 0 0.00 8

5. 0 0.00 16

5. 0 0.00 32

5. 0 0.00 64(OF

)

25. 3/32 1

1 a redundant binary euclidean gcd algorithm s. n. parikh, d. w. matula computer arithmetic 10th...

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