modular arithmetic n.
Download
Skip this Video
Download Presentation
Modular Arithmetic

Loading in 2 Seconds...

play fullscreen
1 / 11

Modular Arithmetic - PowerPoint PPT Presentation


  • 164 Views
  • Uploaded on

Modular Arithmetic. Shirley Moore CS4390/5390 Fall 2013 http://svmoore.pbworks.com/ September 5, 2013 . Agenda. Intro to Matlab by Rogelio Long (cont.) (15 min) Discuss homework from last class (15 min ) Modular arithmetic ( 20 min ) Fermat’s Little Theorem (20 min)

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Modular Arithmetic' - irina


Download Now 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
modular arithmetic

Modular Arithmetic

Shirley Moore

CS4390/5390 Fall 2013

http://svmoore.pbworks.com/

September 5, 2013

agenda
Agenda
  • Intro to Matlab by Rogelio Long (cont.) (15 min)
  • Discuss homework from last class (15 min)
  • Modular arithmetic (20min)
  • Fermat’s Little Theorem (20 min)
  • Wrap-up and preparation for next class (5 min)
modular arithmetic1
Modular Arithmetic
  • The number X (mod Y) is the remainder when X is divided by Y.
    • For example: 7 (mod 3) is 1 because 7 = 2 * 3 + 1. That is, when you divide 7 by 3, you get a remainder of 1.
    • The "modulo Y" terminology can also be used in the following way: Z = X (mod Y), meaning that Z and X have the same remainder when divided by Y. For example: 7 = 25 (mod 3)because 7 = 2 * 3 + 1 and 25 = 8 * 3 + 1
modular addition
Modular Addition
  • Find the units digit of the sum

2403 + 791 + 688 + 4339

i.e., 2403 + 791 + 688 + 4339 (mod 10)

  • In general, if a, b, c, and d are integers and m is a positive integer such that

a = c (mod m) and b = d (mod m)

then

a + b = c + d (mod m)

Proof:

modular multiplication
Modular Multiplication
  • When you take products of many numbers and you want to find their remainder modulo n, you never need to worry about numbers bigger than the square of n.
  • Pick any two numbers x and y, and look at their remainders (mod 7):

a = x (mod 7)

b= y (mod 7)

  • Compare the remainder modulo 7 of the products xy and ab:

xy (mod 7) with ab (mod 7)

  • For example, try x = 26, y = 80
modular multiplication of many numbers
Modular Multiplication of Many Numbers
  • If we want to multiply many numbers modulon, we can first reduce all numbers to their remainders. Then, we can take any pair of them, multiply and reduce again.
  • For example, suppose we want to find

X = 36 * 53 * 91 * 17 * 22 (mod 29)

  • What is the largest number we have to multiply?
modular exponentiation
Modular Exponentiation
  • Suppose we would like to calculate 1143 (mod 13).
  • The straightforward method would be to multiply 11 by 11, then to multiply the result by 11, and so forth. This would require 42 multiplications.
  • We can save a lot of multiplications if we do the following:
    • First write 43 as a sum of powers of 2:

43 = 32 + 8 + 2 + 1

    • That means that 1143 = 1132 * 118 * 112 * 11 .
  • How many multiplications are required, and what is the largest number we have to multiply?
fermat s little theorem
Fermat’s Little Theorem
  • First stated by Pierre de Fermat in 1640
  • First published proof by Leonhard Euler in 1736
  • Highly useful for simplifying the computation of exponents in modular arithmetic
  • Corollary by Euler serves as the basis for RSA encryption
  • Theorem: If p is a prime number and p does not divide a, then ap-1 = 1 (mod p)
  • Example: p = 5
  • Proof: See

http://www.youtube.com/watch?v=w0ZQvZLx2KA

  • Use FLT to find 3100,000 (mod 53)
use flt to prove a number is composite without factoring it
Use FLT to prove a number is composite without factoring it
  • To prove n is composite, find some a such that a is not a multiple of n and an-1 ≠ 1 (mod n).
  • Is 91 a prime number? Try a = 2.
  • 75 = 1 (mod 6), so is 6 prime?
  • True or False: If bn-1 = 1 (mod n) for all b such that b is not a multiple of n, then n is prime.
modular arithmetic in matlab
Modular Arithmetic in Matlab
  • http://www.mathworks.com/help/symbolic/mupad_ug/modular-arithmetic.html
  • mod
  • mods
  • powermod
preparation for next class
Preparation for Next Class
  • Workon Homework 1 (turn in for grade, due September 12). Ask questions next class.