Strong prime

From Wikipedia, the free encyclopedia

A strong prime is a type of prime number that is sometimes used for certain cryptographic applications such as key generation for RSA keys. A strong prime is a prime number p such that both p + 1 and p - 1 have large prime factors, r and s respectively, and r - 1 and s - 1 also have large prime factors. This makes the factorization of an RSA modulus n = pq, where p and q are strong primes, using Pollard's p-1 algorithm computationally infeasible. For this reason, strong primes are required by the ANSI X9.31 standard for use in generating RSA keys for digital signatures. However, strong primes do not protect against modulus factorisation using newer algorithms such as Lenstra elliptic curve factorization, and given the additional cost of generating strong primes RSA Security do not currently recommend their use in key generation.

In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below. Or to put it algebraically, given a prime number $p n$ , where n is its index in the ordered set of prime numbers, $p_n > {{p_{n - 1} + p_{n + 1}} \over 2}$ . The first few strong primes are

11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 (sequence A051634 in OEIS)

For example, 17 is the seventh prime. The sixth and eighth primes, 13 and 19, add up to 32, and half that is 16. That is less than 17, thus 17 is a strong prime.

Given a twin prime {p, p + 2}, the lesser prime of the two, p, will almost certainly be a strong prime. The only twin prime pairs {p, p + 2} for which p is not a strong prime are {3, 5} and {5, 7}.

It is possible for a prime to be a strong prime both in the cryptographic sense and the number theoretic sense. For the sake of illustration, 439351292910452432574786963588089477522344331 is a strong prime in the number theoretic sense because the arithmetic mean of its two neighboring primes is 62 less. Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588089477522344332 has the large prime factor 864608136454559457049 (and in turn the number one less than that has the large prime factor 105646155480762397). Even using algorithms more advanced than trial by division, these numbers would be difficult to factor by hand. For a modern computer algebra system, these numbers can be factored almost instantaneously. A cryptographically strong prime has to be much larger than this example.

[edit] See also

A computationally large safe prime is likely to be a cryptographically strong prime.

Note that the criteria for determining if a pseudoprime is a strong pseudoprime is by congruences to powers of a base, not by inequality to the arithmetic mean of neighboring pseudoprimes.

When a prime is equal to the mean of its neighboring primes, it's called a balanced prime. When it's less, it's called a weak prime.