Strong prime

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In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.

Definition in cryptography

In cryptography, a prime number $p$ is strong if the following conditions are satisfied.^[1]

$p$ is sufficiently large to be useful in cryptography; typically this requires $p$ to be too large for plausible computational resources to enable a cryptanalyst to factorise products of $p$ multiplied by other strong primes.
$p-1$ has large prime factors. That is, $p=a_{1}q_{1}+1$ for some integer $a_{1}$ and large prime $q_{1}$ .
$q_{1}-1$ has large prime factors. That is, $q_{1}=a_{2}q_{2}+1$ for some integer $a_{2}$ and large prime $q_{2}$ .
$p+1$ has large prime factors. That is, $p=a_{3}q_{3}-1$ for some integer $a_{3}$ and large prime $q_{3}$ .

Sometimes a prime that satisfies a subset of the above conditions is also called strong. In some cases, some additional conditions may be included. For example, $a_{1}=2$ , or $a_{2}=2$ , etc.

Definition in number theory

In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). 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.

In a twin prime pair (p, p + 2) with p > 5, p is always a strong prime, since 3 must divide p − 2 which cannot be prime.

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 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.

Application of strong primes in cryptography

Factoring-based cryptosystems

Some people suggest that in the key generation process in RSA cryptosystems, the modulus $n$ should be chosen as the product of two strong primes. This makes the factorization of $n=pq$ 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 Number Field Sieve algorithm. Given the additional cost of generating strong primes RSA Security do not currently recommend their use in key generation. Similar (and more technical) argument is also given by Rivest and Silverman.^[1]

Discrete-logarithm-based cryptosystems

It is shown by Stephen Pohlig and Martin Hellman in 1978 that if all the factors of p-1 are less than $\log ^{c}p$ , then the problem of solving discrete logarithm modulo p is in P. Therefore, for cryptosystems based on discrete logarithm, such as DSA, it is required that p-1 have at least one large prime factor.

References

↑ 1.0 1.1 Ron Rivest and Robert Silverman, Are 'Strong' Primes Needed for RSA?, Cryptology ePrint Archive: Report 2001/007. http://eprint.iacr.org/2001/007

External links

Guide to Cryptography and Standards
3.1.4 What are Strong Primes and are they Necessary for the RSA System? - RSA Lab's explanation on strong vs weak primes

v t e Prime number classes

By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))

By integer sequence	Fibonacci Lucas Motzkin Bell Partitions Pell Perrin Newman–Shanks–Williams

By property	Lucky Wall–Sun–Sun Wilson Wieferich Wieferich pair Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Wolstenholme Good Super Higgs Highly cototient Illegal

Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Full reptend Unique Primeval Self Smarandache–Wellin

Patterns	Twin (p, p + 2) Bi-twin chain (p − 1, p + 1, 2p − 1, 2p + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)

By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known

Complex numbers	Eisenstein prime Gaussian prime

Composite numbers	Pseudoprime Almost prime Semiprime Interprime

Related topics	Probable prime Industrial-grade prime Formula for primes Prime gap

List of prime numbers

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