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This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License . CS 312: Algorithm Analysis. Lecture #4: Primality Testing, GCD. Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick. Announcements. HW #2 Due Now

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cs 312 algorithm analysis

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.

CS 312: Algorithm Analysis

Lecture #4: Primality Testing, GCD

Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

announcements
Announcements
  • HW #2 Due Now
  • Homework: Required to show your work
  • Project #1
    • Today we’ll work through the rest of the math
    • Early: this Wednesday, 1/16
    • Due: Friday, 1/18
  • Bonus help session on Proof by Induction
    • Tuesday, 4pm, 1066 TMCB
practice
Practice
  • Key points:
  • Represent exponent in binary
  • Break up the problem into factors (one per binary digit)
  • Use the substitution rule
objectives
Objectives
  • Part 1:
    • Introduce Fermat’s Little Theorem
    • Understand and analyze the Fermat primality tester
  • Part 2:
    • Discuss GCD and Multiplicative Inverses, modulo N
    • Prepare to Introduce Public Key Cryptography
  • This adds up to a lot of ideas!
fermat s little theorem
Fermat’s Little Theorem

If p is prime, then a p-1  1 (mod p)

for any a such that

Examples:

How do you wish you could use this theorem?

p = 3, a = 2

p = 7, a = 4

logic review
Logic Review

a  b (a implies b)

Which is equivalent to the above statement?

  • b  a
  • ~a  ~b
  • ~b  ~a
logic review1
Logic Review

a  b (a implies b)

Which is equivalent to the above statement?

  • b  a The Converse
  • ~a  ~b The Inverse
  • ~b  ~a The Contrapositive
contrapositive of fermat s little theorem
Contrapositive of Fermat’s Little Theorem

If pand a are integers such that and a p-1 mod p 1, then p is not prime.

first prime number test
First Prime Number Test

function primality(N)

Input: Positive integer N

Output: yes/no

// a is random positive integer between 1 and N-1

a = uniform(1..N-1)

//

if (modexp(a, N-1, N) == 1):

return “possibly prime”

else: return “not prime” // certain

false witnesses
False Witnesses
  • If primality(N) returns “possibly prime”, then N might or might not be prime, as the answer indicates
  • Consider 15:
    • 414 mod 15 = 1
    • but 15 clearly isn’t prime!
    • 4 is called a false witness of 15
  • Given a non-prime N, we call a number a where an-1 mod N = 1 a “false witness” (to the claim that N is prime)
relatively prime
Relatively Prime
  • Two numbers, a and N, are relatively prime if their greatest common divisor is 1.
    • 3 and 5?
    • 4 and 8?
    • 4 and 9?
  • Consider the Carmichael numbers:
    • They pass the test (i.e., an-1 mod N = 1) for all a relatively prime to N.
false witnesses1
False Witnesses
  • Ignoring Carmichael numbers,
  • How common are false witnesses?
    • Lemma: If an-1 mod n = 1 for some a relatively prime to n, then it must hold for at least half the choices of a < n
state of affairs
State of Affairs
  • Summary:
    • If n is prime, then an-1 mod n = 1 for all a < n
    • If n is not prime, then an-1 mod n = 1 for at most half the values of a < n
  • Allows us to put a bound on how often our primality() function is wrong.
correctness
Correctness
  • Question #1: Is the “Fermat test” correct?
    • No
  • Question #1’: How correct is the Fermat test?
    • The algorithm is ½-correct with one-sided error.
      • The algorithm has 0.5 probability of saying “yes N is prime” when N is not prime.
      • But when the algorithm says “no N is not prime”, then N must not be prime (by contrapositive of Fermat\'s Little Theorem)
amplification
Amplification
  • Repeat the test
    • Decrease the probability of error:

C = Composite; P = Prime

  • Amplification of stochastic advantage
p correct
P(Correct)
  • k trials gives 1/(2k) probability of being incorrect when the answer is "prime“

P(Correct) =

modified primality test
Modified Primality Test

function primality2(N)

Input: Positive integer N

Output: yes/no

for i = 1 to k do:

a = uniform(1..N-1)

if (modexp(a, N-1, N) == 1):

// possibly prime; do nothing

else: return “not prime”

return yes

greatest common divisor
Greatest Common Divisor
  • Euclid’s rule:
    • gcd(x, y) = gcd (x mod y, y) = gcd(y, x mod y)
  • Can compute gcd(x,y) for large x, y by modular reduction until we reach the base case!

function Euclid (a,b)

Input: Two integers a and b with a  b  0 (n-bit integers)

Output: gcd(a,b)

if b=0: return a

return Euclid(b, a mod b)

example
Example
  • gcd(25, 11)
3 questions
3 Questions
  • 1. Is it Correct?
  • 2. How long does it take?
  • 3. Can we do better?
analysis
Analysis

function Euclid (a,b)

Input: Two integers a and b with a  b  0 (n-bit integers)

Output: gcd(a,b)

if b=0: return a

return Euclid(b, a mod b)

bezout s identity
Bezout’s Identity
  • For two integers a, band their GCD d, there exist integers x and y such that:
extended euclid algorithm
Extended Euclid Algorithm

function extended-Euclid (a, b)

Input: Two positive integers a & b with a  b  0 (n-bits)

Output: Integers x, y, d such that d = gcd(a, b)and ax + by = d

if b=0: return (1,0,a)

(x’, y’, d) = extended-Euclid(b, a mod b)

return (y’, x’ – floor(a/b)y’, d)

example1
Example
  • Note: there’s a great worked example of how to use the extended-Euclid algorithm on Wikipedia here:

http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm

  • And another linked from the reading column on the schedule for today
multiplicative inverses
Multiplicative Inverses
  • In the rationals, what’s the multiplicative inverse of a?
  • In modular arithmetic (modulo N), what is the multiplicative inverse?
multiplicative inverses1
Multiplicative Inverses
  • In the rationals, what’s the multiplicative inverse of a?
  • In modular arithmetic (modulo N), what is the multiplicative inverse?
finding multiplicative inverses modulo n
Finding Multiplicative Inverses(Modulo N)
  • Modular division theorem: For any a mod N, a has a multiplicative inverse modulo N if and only if it is relatively prime to N.
  • Significance of extended-Euclid algorithm:
    • When two numbers, and , are relatively prime
    • extended-Euclid algorithm produces and such that
    • Thus,
      • Because for all integers
    • Then is the multiplicative inverse of modulo
  • i.e., I can use extended-Euclid to compute the multiplicative inverse of mod
multiplicative inverses2
Multiplicative Inverses
  • The multiplicative inverse mod is exactly what we will need for RSA key generation
  • Notice also: when and are relatively prime, we can perform modular division in this way
slide31
Next
  • RSA
assignment
Assignment
  • Read and Understand: Section 1.4
  • HW #3: 1.9, 1.18, 1.20
  • Finish your project #1 early!
    • Submit on Wednesday by 5pm
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