Pseudoprimes 1 / 9 Two Related Problems How do we determine whether a given integer is prime? How do we determine whether a given integer is composite? 2 / 9 Pseudoprimes A pseudoprime is an odd composite integer that shares some of the properties of actual primes. Definition An integer n is called a pseudoprime to the base b when n is odd and composite with gcd(b; n) = 1 and bn−1 ≡ 1 (mod n). 3 / 9 The Number of Bases for which n is a Pseudoprime Proposition Let n be an odd composite integer. (a) An odd composite number n is a pseudoprime to the base b if and only if gcd(b; n) = 1 and the order of b in (Z=nZ)∗ divides n − 1. (b) If n is a pseudoprime to the bases b1 and b2, then n is a −1 pseudoprime to the base b1b2 and also to the base b1b2 −1 (where b2 is an integer which is multiplicatively inverse to b2 modulo n). (c) If bn−1 6≡ 1 (mod n) for some b relatively prime to n, then bn−1 6≡ 1 (mod n) for at least half of the possible bases b 2 (Z=nZ)∗. 4 / 9 unless n is an integer such that bn−1 ≡ 1 (mod n) for all b relatively prime to n. Probable Primes A consequence of the last proposition is that the probability of an integer n being composite with bases b1;:::; bk relatively n−1 prime to n such that bi ≡ 1 (mod n) for 1 ≤ i ≤ k 1 is at most 2k 5 / 9 Probable Primes A consequence of the last proposition is that the probability of an integer n being composite with bases b1;:::; bk relatively n−1 prime to n such that bi ≡ 1 (mod n) for 1 ≤ i ≤ k 1 is at most 2k unless n is an integer such that bn−1 ≡ 1 (mod n) for all b relatively prime to n. 6 / 9 Carmichael Numbers Definition A Carmichael number is a composite integer n such that bn−1 ≡ 1 (mod n) for every b relatively prime to n. Note The smallest Carmichael number is 561. 7 / 9 Properties of Carmichael Numbers Proposition Let n be an odd composite integer. (a) If n is divisible by a perfect square greater than 1, then n is not a Carmichael number. (b) If n is square free, then n is a Carmichael number if and only if (p − 1)j(n − 1) for every prime p dividing n. Proposition A Carmichael number must be the product of at least three distinct primes. 8 / 9 Strong Pseudoprimes Definition Let n be an odd composite integer with n − 1 = 2s t where t is odd and let b be an integer relatively prime to n. We say that n is a strong pseudoprime to the base b when either bt ≡ 1 (mod n) or there exists r with 0 ≤ r < s such that b2r t ≡ −1 (mod n). Proposition If n is an odd composite integer, then at least 75% of the bases b in the range 0 < b < n act as Rabin-Miller witnesses for n. 9 / 9.