1 / 15

1.5.2 The Algorithm of Cornacchia

1.5.2 The Algorithm of Cornacchia. 今回の内容. Algorithm1.5.2(Cornacchia) Algorithm1.5.2 の実装 Algorithm1.5.2 の Example Algorithm1.5.3(Modified-Cornacchia). Algorithm1.5.2(Cornacchia). より、一般的に・・・・. Algorithm1.5.2(Cornacchia). Input: Output: Step1: Step2:. Algorithm1.5.2(Cornacchia). Step3:

gallia
Télécharger la présentation

1.5.2 The Algorithm of Cornacchia

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1.5.2The Algorithm of Cornacchia

  2. 今回の内容 • Algorithm1.5.2(Cornacchia) • Algorithm1.5.2の実装 • Algorithm1.5.2のExample • Algorithm1.5.3(Modified-Cornacchia)

  3. Algorithm1.5.2(Cornacchia) より、一般的に・・・・

  4. Algorithm1.5.2(Cornacchia) • Input: • Output: • Step1: • Step2:

  5. Algorithm1.5.2(Cornacchia) • Step3: • Step4: このbがxの候補になる Euclid Algorithmの適用

  6. Algorithm1.5.2(Cornacchia) • 証明について • F. Morain, J.-L. Nicolas による証明がある. (URL) http://web.math.hr/~duje/tbkript/tbksem.html • について • Primes of the form x2+ny2 (Cox, David A. 著) に多くのことが書かれている.

  7. Pythonでの実装 import math import kro1 import shanks import square_test def cornacchia(d,p): k = kro1.kro_b(-d,p) if k == -1: return "no solution" x0= shanks.shanks(-d,p) if x0 == 0: return "no solution" if x0 < p/2: x0 = p – x0 a = p b = x0 l = int(math.floor(math.sqrt(p))) while b > l: r = a % b a = b b = r c = (p - (b ** 2)) / d t = squaretest.square_test(c) if ((d % (p - (b ** 2)) != 0) or square_test.square_test(c) > 1): return (b,t) return "no solution" 平方剰余の計算 Algorithm1.5.1 Algorithm1.7.3

  8. Pythonでの実装(Algorithm1.5.1) k = kro1.kro_b(n,g) z = (n ** q) % g y = z r = e x = (c ** ((q - 1) // 2)) % g b = (c * (x ** 2)) % g x = (c * x) % g while b % g != 1: m = 1 while ((b ** (2 ** m)) % g) != 1: m = m + 1 if m == r: print “[a] is not a quadratic residue mod p" return 0 t = (y ** (2 ** (r - m - 1))) % g y = (t ** 2) % g r = m % g x = (x * t) % g b = (b * y) % g return x import math import random import kro1 def shanks(c,g): temp = g -1 e = 0 while temp % 2 == 0: temp = temp // 2 e = e + 1 q = (g-1) // 2**e k = 0 while k != -1: n =int(math.floor (1000*random.random()))

  9. Pythonでの実装(Algorithm1.7.1&3) for k in range(0,64): q64.append(0) for k in range(0,32): q64[(k ** 2) % 64] = 1 for k in range(0,65): q65.append(0) for k in range(0,33): q65[(k ** 2) % 65] = 1 t = c % 64 if q64[t] == 0: return "sono1" r = c % 45045 if q63[r % 63] == 0: return "sono2" if q65[r % 65] == 0: return "sono3" if q11[r % 11] == 0: return "sono4" x = c y = 0 y = math.floor ((x + math.floor(c/x))/2) while y < x: x = y y = math.floor ((x + math.floor(c/x))/2) if int(math.floor(x ** 2)) == int(math.floor(c)): return 1 else: return 0 import math def square_test(c): q11 = [] q63 = [] q64 = [] q65 = [] for k in range(0,11): q11.append(0) for k in range(0,6): q11[(k ** 2) % 11] =1 for k in range(0,63): q63.append(0) for k in range(0,32): q63[(k ** 2) % 63] = 1

  10. Example p/2<x0<p となるように • Step1: • Step2: • Step3: a b r 97 80 - 80 17 5 17 12 5 12 5 5 • Step4: Euclid Algorithm を適用 平方数 になっている

  11. Algorithm1.5.3(Modified Cornacchia) (1) (2) (3)

  12. Algorithm1.5.3(Modified Cornacchia) • (3)の場合,Algorithm1.5.2を適用することができない • 1.5.2を修正したAlgorithm1.5.3を用いることで,この問題を解決することができる のとき

  13. Algorithm1.5.3(Modified Cornacchia) • Input: • Output: • Step1: • Step2: 1.5.2との違い

  14. Algorithm1.5.3(Modified Cornacchia) • Step3: • Step4: 1.5.2との違い

  15. Algorithm1.5.3(Modified Cornacchia) • Step5:

More Related