Number-Theoretic Algorithms

Number-Theoretic Algorithms PowerPoint PPT Presentation


  • 150 Views
  • Uploaded on
  • Presentation posted in: General

. Prime Number: 2, 3, 5, 7, 11, 13

Download Presentation

Number-Theoretic Algorithms

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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


1. Number-Theoretic Algorithms

2. Prime Number: 2, 3, 5, 7, 11, 13…. Composite number: 4, 6, 8,…..

3. Thm1: (Division Theorem) For any integer a and any positive integer n, there are unique integers q and r such that 0?r<n and a=qn+r. q=?a/n? is the quotient of the division. r= a mod n is the remainder of the division. a =?a/n?n+(a mod n) Zn={0,1,2,…,n-1}

4. Greatest Common Divisor(GCD) Gcd(24,30)=6, gcd(5,7)=1 Thm2: If a and b are any integers, not both zero, then gcd(a,b) is the smallest positive element of the set {ax+by:x,y?Z} of linear combinations of a and b.

5. Pf: Let s be the smallest positive ax+by for some x,y?Z. a mod s = a- ?a/s?s =a-?a/s?(ax+by) =a(1- ?a/s?x)+b(-?a/s?y).

6. a mod s = a(1- ?a/s?x)+b(-?a/s?y). Thus (a mod s) is a linear combination of a and b. ?a mod s<s ? a mod s =0 because s is the smallest positive such linear combination. ? s|a.

7. Similarly, s|b ? s is a common divisor of a and b. ?gcd(a,b)?s. ? s=ax+by ? gcd(a,b) | s ?gcd(a,b)?s. Thus, gcd(a,b)=s.

  • Login