Euclid & GCD algorithm

Description:

• def EuclidAgl(a,b):
      if b = 0:
    	  return a
      else:
    	  return EuclidAlg(b, a mod b)

  • If $a,b\in \mathbb{N},b\ne 0$ then $gcd(a,b)=gcd(b,a\mathrm{mod}b)$lemma
• Euclid’s algorithm, on input EuclidAlg(a, b) for $a,b\in N^{+},a,b$ not both 0, always terminates and produces the correct output.theorem
• For every two recursive calls made by EuclidAlg, the first argument a is at least reduced by halved.claim
• theorem Euclid’s algorithm, on input EuclidAlg(a, b) for $a,b\in N+$, a, b not both 0, always terminates in time proportional to $lo{g}_{2}a$
  • if a larger, algorithm increase logarithmically, not linearly

Theorem 11 (Bézout Theorem)

• Let a, b ∈ N+, a, b not both 0. Then, there exist s, t (Bézout coefficients) ∈ Z such that $sa+tb=gcd(a,b)$ (Bézout identity)
• How to compute $s,t$ from $gcd(a,b)=z$
  • write out a = b.q + r for each recursive call (from $r_1$ → $r_n$ , for eg) of solving EuclidAlg(a, b), until there is no remainder left
  • writing in backward order, from z = elements of $r_1$ → $r_n$
  • compute so that you get the form $sa+tb=z$