Download presentation

Presentation is loading. Please wait.

Published byTamsin Howard Modified over 2 years ago

1
**COM 5336 Cryptography Lecture 7a Primality Testing**

Scott CH Huang 1 1

2
Definition A prime number is a positive integer p having exactly two positive divisors, 1 and p. A composite number is a positive integer n > 1 which is not prime. COM 5336 Cryptography Lecture 7a

3
**Primality Test vs Factorization**

Factorization’s outputs are non-trivial divisors. Primality test’s output is binary: either PRIME or COMPOSITE COM 5336 Cryptography Lecture 7a

4
**Naïve Primality Test Input: Integer n > 2**

Output: PRIME or COMPOSITE for (i from 2 to n-1){ if (i divides n) return COMPOSITE; } return PRIME; COM 5336 Cryptography Lecture 7a

5
**Still Naïve Primality Test**

Input/Output: same as the naïve test for (i from 1 to ){ if (i divides n) return COMPOSITE; } return PRIME; COM 5336 Cryptography Lecture 7a

6
**Sieve of Eratosthenes Input/Output: same as the naïve test**

Let A be an arry of length n Set all but the first element of A to TRUE for (i from 2 to ){ if (A[i]=TRUE) Set all multiples of i to FALSE } if (A[i]=TRUE) return PRIME else return COMPOSITE COM 5336 Cryptography Lecture 7a

7
**Primality Testing Two categories of primality tests Probablistic**

Miller-Rabin Probabilistic Primality Test Cyclotomic Probabilistic Primality Test Elliptic Curve Probabilistic Primality Test Deterministic Miller-Rabin Deterministic Primality Test Cyclotomic Deterministic Primality Test Agrawal-Kayal-Saxena (AKS) Primality Test COM 5336 Cryptography Lecture 7a

8
**Running Time of Primality Tests**

Miller-Rabin Primality Test Polynomial Time Cyclotomic Primality Test Exponential Time, but almost poly-time Elliptic Curve Primality Test Don’t know. Hard to Estimate, but looks like poly-time. AKS Primality Test Poly-time, but only asymptotically good. COM 5336 Cryptography Lecture 7a

9
**Fermat’s Primality Test**

It’s more of a “compositeness test” than a primality test. Fermat’s Little Theorem: If p is prime and , then If we can find an a s.t. , then n must be a composite. If, for some a, n passes the test, we cannot conclude n is prime. Such n is a pseudoprime. If this pseudoprime n is not prime, then this a is called a Fermat liar. If, for all we have can we conclude n is prime? No. Such n is called a Carmichael number. n is a Fermat liar if it passes the test and still is composite a is a Fermat witness if it identifies a composite Carmichael Numbers: 561 ( ), 1105( ),1729( ) What is required to write the fermats algorithm COM 5336 Cryptography Lecture 7a

10
**Pseudocode of Fermat’s Primality Test**

FERMAT(n,t){ INPUT: odd integer n ≥ 3, # of repetition t OUTPUT: PRIME or COMPOSITE for (i from 1 to t){ Choose a random integer a s.t. 2 ≤ a ≤ n-2 Compute if ( ) return COMPOSITE } return PRIME COM 5336 Cryptography Lecture 7a

11
**Miller-Rabin Probabilistic Primality Test**

It’s more of a “compositeness test” than a primality test. It does not give proof that a number n is prime, it only tells us that, with high probability, n is prime. It’s a randomized algorithm of Las Vegas type. COM 5336 Cryptography Lecture 7a

12
**A Motivating Observation**

FACT: Let p be an odd prime If then Moreover, if , and m is odd. Let Then either or for some COM 5336 Cryptography Lecture 7a

13
Miller-Rabin If and Then a is a strong witness for the compositeness of n. If or for some then n is called a pseudoprime w.r.t. base a, and a is called a strong liar. COM 5336 Cryptography Lecture 7a

14
**Miller-Rabin: Algorithm Pseudocode**

MILLER-RABIN(n,t){ INPUT: odd integer n ≥ 3, # of repetition t Compute k & odd m s.t. for ( i from 1 to t ){ Choose a random integer a s.t. Compute if ( or ){ Set while( and ){ if ( ) return COMPOSITE } if ( ) return COMPOSITE return PRIME COM 5336 Cryptography Lecture 7a

15
**Miller-Rabin: Example**

90 = 2*45, s = 1, r = 45 {1,9,10,12,16,17,22,29,38,53,62,69,74, 75,79,81,82,90} are all strong liars. 945 = 1 mod 91 1045 = 1 mod 91 …. All other bases are strong witnesses. 97 = 9 mod 91 98 = 81 mod 91 COM 5336 Cryptography Lecture 7a

16
**Miller-Rabin: Main Theorem**

Given n > 9. Let B be the number of strong liars. Then If the Generalized Riemann Hypothesis is true, then Miller-Rabin primality test can be made deterministic by running MILLER-RABIN(n, 2log2n) COM 5336 Cryptography Lecture 7a

Similar presentations

OK

Lecture 8: Primality Testing and Factoring Piotr Faliszewski

Lecture 8: Primality Testing and Factoring Piotr Faliszewski

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on team building training module Topics for ppt on environmental ethics Ppt on central excise duty Ppt on network theory communication Ppt on social networking sites project Download ppt on measures of dispersion Ppt on c language operators Ppt on network security issues Download ppt on phase controlled rectifier Ppt on dispersal of seeds by animals puzzle