mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a strong prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

with certain special properties. The definitions of strong primes are different in

cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...

and

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

Definition in number theory

, a strong prime is a prime number that is greater than the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of the nearest prime above and below (in other words, it is closer to the following than to the preceding prime). Or to put it algebraically, writing the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of prime numbers as (''p'', ''p'', ''p'', ...) = (2, 3, 5, ...), ''p'' is a strong prime if . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime. The first few strong primes are : 11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101,

107 107 may refer to: *107 (number), the number *AD 107, a year in the 2nd century AD *107 BC, a year in the 2nd century BC *107 (New Jersey bus) *107 Camilla, a main-belt asteroid *Peugeot 107, a city car See also

*10/7 (disambiguation) *Bohrium, ...

, 127, 137, 149,

163 Year 163 ( CLXIII) was a common year starting on Friday of the Julian calendar. At the time, it was known as the Year of the Consulship of Laelianus and Pastor (or, less frequently, year 916 ''Ab urbe condita''). The denomination 163 for this y ...

, 179, 191, 197, 223, 227, 239,

251 __NOTOC__ Year 251 (Roman numerals, CCLI) was a common year starting on Wednesday of the Julian calendar. At the time, in the Roman Empire, it was known as the Year of the Consulship of Traianus and Etruscus (or, less frequently, year 1004 ''A ...

, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 . In a

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''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.

Definition in cryptography

, a prime number ''p'' is said to be "strong" if the following conditions are satisfied. * ''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'' with other strong primes. * ''p'' − 1 has large prime factors. That is, ''p'' = ''a'q'' + 1 for some integer ''a'' and large prime ''q''. * ''q'' − 1 has large prime factors. That is, ''q'' = ''a'q'' + 1 for some integer ''a'' and large prime ''q''. * ''p'' + 1 has large prime factors. That is, ''p'' = ''a'q'' − 1 for some integer ''a'' and large prime ''q''. 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 A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...

, 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 RSA Security LLC, formerly RSA Security, Inc. and trade name RSA, is an American computer security, computer and network security company with a focus on encryption and decryption standards. RSA was named after the initials of its co-founders, ...

do not currently recommend their use in key generation. Similar (and more technical) argument is also given by Rivest and Silverman.

Discrete-logarithm-based cryptosystems

It is shown by Stephen Pohlig and

Martin Hellman Martin Edward Hellman (born October 2, 1945) is an American cryptologist and mathematician, best known for his invention of public-key cryptography in cooperation with Whitfield Diffie and Ralph Merkle. Hellman is a longtime contributor to the ...

in 1978 that if all the factors of ''p'' − 1 are less than log ''p'', then the problem of solving

discrete logarithm In mathematics, for given real numbers a and b, the logarithm \log_b(a) is a number x such that b^x=a. Analogously, in any group G, powers b^k can be defined for all integers k, and the discrete logarithm \log_b(a) is an integer k such that b^k=a ...

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.

Miscellaneous facts

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 is called a balanced prime. When it is less, it is called a weak prime (not to be confused with a weakly prime number).

References

External links

Guide to Cryptography and Standards

- RSA Lab's explanation on strong vs weak primes {{Prime number classes Classes of prime numbers Theory of cryptography