Carmichael Numbersand

Primality Tests

By

Sanjeev Rao

Outline Introduction Carmichael numbers

What is Carmichael number Detecting a Carmichael number Statistics and importance

Ancient Primality testing methods Sieve of Eratosthenes Chinese Primality Test

Introduction Cryptographic algorithms uses big

prime numbers Checking a big number is prime is not

so easy

Solution Use probabilistic primality tests Fermat Little Theorem:

If n is prime then

ap-1 ≡ 1 (mod p) for any integer a and p a∤ .

Carmichael Numbers Pseudoprime for every possible base b: that

is, for every b coprime to n Passes Fermats little theorem test

Statistics 16 #s up to 100,000 43 #s up to 106

105,212 up to 1015 and 246,683 up to 1016

Example – 561, 1105, 1729, 2465 ….

Detecting Carmichael numbers

If n is a product of distinct prime numbers,n = p1, p2, p3 ……. Ps , pi ≠ pj and

pi –1 | n-1 for every prime factor pi, i = 1…….. s,

then n is a Carmichael number.

Example: n = 561 = 3 x 11 x 17 2 | 560 , 10 | 560, 16 | 560

Importance Encryption algorithms like RSA,

ElGamal etc must have large primes For example, If we pick a Carmichael

number as a prime number p in RSA, we can factor p and hence q and k

( k = (p-1) x (q-1) )

Primality testing Process of proving a number is

prime Two of the oldest test methods

Sieve of Eratosthenes Chinese Primality Test

Sieve of Eratosthenes Greek mathematician Found this method in 240 BC One of the most efficient way to find all of the small

primes (say all those less than 10,000,000) Sieve all primes less than given n

Sieve of Eratosthenes contd…

Write down the numbers 1, 2, 3, ..., n. We will eliminate composites by marking them. Initially all numbers are unmarked

Mark the number 1 as special (it is neither prime nor composite)

Sieve of Eratosthenes contd… Set k=1. Until k exceeds or equals the square root of n

do this: Find the first number in the list greater than k that has

not been identified as composite. (The very first number is 2.) Call it m. Mark the numbers 2m, 3m, 4m, ... as composite. (Thus in the first run we mark all even numbers greater than 2. In the second run we mark all multiples of 3 greater than 3.)

m is a prime number. Put it on your list Set k=m and repeat Put the remaining unmarked numbers in the sequence

on your list of prime numbers

Example primes less than or equal to 30

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

The first number 2 is prime, cross out its multiples (color them red), so the red numbers are not prime.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Example contd…

Repeat this with the next number 3 and so on…

Finally we have2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 30 We are left with {2, 3, 5, 7, 11, 13, 17,

19, 23, 29} as primes less than 30

contd…

speed O(n(log n)log log n) bit operations

space O(n)

Disadvantages Not efficient for big numbers Needs lot of space to store big

numbers

Possible solution Large n use a segmented sieve http://www.ieeta.pt/~tos/software/prime_sieve.html

Gives the algorithm and the runtime in seconds on a 900MHz Athlon processor with 512Mbytes of memory running on GNU/Linux

A1, A2, A3 three different segmented sieve algorithms

Performance

n A1 A2 A3

12 13.31 14.08 2.63

14 25.79 20.62 4.05

16 97.39 26.26 5.58

18 208.13

32.36 9.61

Chinese Primality Test Found in approximately 500 B.C Let n be an integer, n > 1.

If 2n is congruent to 2 (mod n) or 2n-1 ≡ 1 (mod n) ,

then n is either a prime or a base-2 pseudoprime.

A number that passes the Chinese Primality Test has only a 0.002% chance of not being prime.

Contd … In 1640, Fermat rediscovered what the

ancient Chinese had known nearly 2000 years before him.

He also examined the problem using bases other than 2, improving on the accuracy of the Chinese test.

References Definitions http://mathworld.wolfram.com/CarmichaelNumber.html

Sieve of Eratostheneshttp://primes.utm.edu/glossary/page.php?sort=SieveOfEratostheneshttp://ccins.camosun.bc.ca/~jbritton/jberatosthenes.htmhttp://www.math.pku.edu.cn/stu/eresource/wsxy/sxrjjc/wk/Encyclopedia/math/e/

e232.htm

http://www-2.cs.cmu.edu/afs/cs/project/pscico/doc/nesl/manual/node10.html Segmented Sievehttp://www.ieeta.pt/~tos/software/prime_sieve.html

References contd… Chinese Primality testinghttp://www-math.mit.edu/phase2/UJM/vol1/DORSEY-F.PDF List of Carmichael Numbers – http://www.kobepharma-u.ac.jp/~math/note/note02.txt

Questions

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