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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:

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1.5.2 The Algorithm of Cornacchia

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  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:

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