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報告者： 張書豪

GCD The Design and Analysis of Computer Algorithms 8.8 Greatest common divisors and Euclid’s algorithm. 報告者： 張書豪. Outline. Half GCD Example. HGCD Algorithm. procedure HGCD(a 0 ,a 1 ) ： If DEG(a 1 ) ≦DEG(a 0 )/2 then else begin

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報告者： 張書豪

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  1. GCDThe Design and Analysis of Computer Algorithms 8.8 Greatest common divisors and Euclid’s algorithm 報告者：張書豪

  2. Outline • Half GCD • Example

  3. HGCD Algorithm procedure HGCD(a0,a1)： If DEG(a1) ≦DEG(a0)/2 then else begin let a0=b0xm+c0, where m= and DEG(c0) ＜m； let a1＝b1xm＋c1,where DEG(c1)<m； R←HGCD(b0,b1)；

  4. f ← d modulo e； end

  5. Example HGCD(a0,a1) a0(x)＝x5+x4+x3+x2+x+1 a1(x)＝x4-2x3+3x2-x-7 由a0=b0xm+c0與a1＝b1xm＋c1,可得到 b0= x3+x2+x+1, c0=x+1, b1=x2-2x+3, c1=-x-7, HGCD(x3+x2+x+1, x2-2x+3) DEG(b1) ＞DEG(b0)/2

  6. 再做HGCD(b0,b1) a0(x)＝ x3+x2+x+1, a1(x)＝x2-2x+3, 由a0=(x2+x+1)x1+1, a1＝(x-2)x1+3,可得到 b0= x2+x+1, c0=1, b1=x-2, c1=3, 此時DEG(b1) ＝DEG(b0)/2 R←HGCD(b0,b1)

  7. 可得d= x3+x2+x+1,e= x2-2x+3,f=d mod e=4x-8 m/2=3/2下限為1,e=(x-2)x+3,f=4(x)-8, g0=x-2,g1=4x0,DEG(0)=DEG(1)/2的下限 S←HGCD(g0,g1)

  8. R←HGCD(b0,b1) d= x4-2x3+3x2-x-7 e= 4x3-7x2+11x+22 f ← d modulo e；

  9. e=(4x2-7x+11)x1+22,

  10. b0=4x3-7x2+11x+22, b1=-3/16x2-93/16x-45/8, DEG(2)>DEG(3)/2下限 →b0=(4x2-7x+11)x+22 b1=(-3/16x-93/16)x-45/8 DEG(1)=DEG(2)/2

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