Carmichael Numbers and Primality Tests By Sanjeev Rao

Outline n n Introduction Carmichael numbers n n What is Carmichael number Detecting a Carmichael number Statistics and importance Ancient Primality testing methods n n Sieve of Eratosthenes Chinese Primality Test

Introduction n n Cryptographic algorithms uses big prime numbers Checking a big number is prime is not so easy

Solution n n 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 n n Pseudoprime for every possible base b: that is, for every b coprime to n Passes Fermats little theorem test

Statistics n n 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 = p 1, p 2, p 3 ……. 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 n n Encryption algorithms like RSA, El. Gamal 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 n n Process of proving a number is prime Two of the oldest test methods n n Sieve of Eratosthenes Chinese Primality Test

Sieve of Eratosthenes n n 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) Sieve all primes less than given n

Sieve of Eratosthenes n n 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 n n n 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 2 m, 3 m, 4 m, . . . 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 n n 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 n n contd… Repeat this with the next number 3 and so on… Finally we have 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 n We are left with {2, 3, 5, 7, 11, 13, 17, 19, 23, 29} as primes less than 30

contd… n n speed O(n(log n)log n) bit operations space O(n)

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

Possible solution n Large n use a segmented sieve n http: //www. ieeta. pt/~tos/software/prime_sieve. html Gives the algorithm and the runtime in seconds on a 900 MHz Athlon processor with 512 Mbytes of memory running on GNU/Linux A 1, A 2, A 3 three different segmented sieve algorithms

Performance n 12 14 16 18 A 1 13. 31 25. 79 97. 39 208. 13 A 2 14. 08 20. 62 26. 26 32. 36 A 3 2. 63 4. 05 5. 58 9. 61

Chinese Primality Test n n n Found in approximately 500 B. C Let n be an integer, n > 1. If 2 n is congruent to 2 (mod n) or 2 n-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 … n n 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 n Definitions http: //mathworld. wolfram. com/Carmichael. Number. html n Sieve of Eratosthenes http: //primes. utm. edu/glossary/page. php? sort=Sieve. Of. Eratosthenes http: //ccins. camosun. bc. ca/~jbritton/jberatosthenes. htm http: //www. math. pku. edu. cn/stu/eresource/wsxy/sxrjjc/wk/Encyclopedia/math/e/e 232. htm http: //www-2. cs. cmu. edu/afs/cs/project/pscico/doc/nesl/manual/node 10. html n Segmented Sieve http: //www. ieeta. pt/~tos/software/prime_sieve. html

References contd… Chinese Primality testing http: //www-math. mit. edu/phase 2/UJM/vol 1/DORSEY-F. PDF n List of Carmichael Numbers – http: //www. kobepharma-u. ac. jp/~math/note 02. txt n

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