Strong prime

Key takeaways

Strong prime belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Strong prime to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Strong prime from memory before moving on to harder problems.

Reference excerpt

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 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 is closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p1, p2, p3, ...) = (2, 3, 5, ...), pn is a strong prime if pn > ⁠pn − 1 + pn + 1/2⁠. 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, 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 the OEIS). 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.

Definition in cryptography 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 = a1q1 + 1 for some integer a1 and large prime q1. q1 − 1 has large prime factors. That is, q1 = a2q2 + 1 for some integer a2 and large prime q2. p + 1 has large prime factors. That is, p = a3q3 − 1 for some integer a3 and large prime q3.

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.

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 logc 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.

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

Worked examples

Example 1 — a first encounter with Strong prime

Start with the simplest possible case. Write down what Strong prime claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Strong prime before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Strong prime ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Strong prime

In research: Strong prime appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Strong prime in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Strong prime is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classes of prime numbers, Theory of cryptography, so understanding it makes those chapters shorter.
In everyday life: Look for Strong prime outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Strong prime in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Strong prime means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Strong prime out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Strong prime in simple terms?

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.

Why does Strong prime matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Strong prime?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Strong prime.