multipleprime

Multipleprime, sometimes written multiprime, is a term used in number theory and cryptography to describe a positive integer that can be expressed as a product of at least three prime factors. In general, primes may be repeated in the factorization unless a stricter variation specifies distinct primes.

Formally, let Omega(n) denote the total number of prime factors of n counted with multiplicity, and let omega(n) denote the number of distinct prime factors. A positive integer n is a multipleprime if Omega(n) ≥ 3. If the primes in the factorization are required to be distinct, the corresponding condition is omega(n) ≥ 3.

Examples are common and straightforward. The number 30 equals 2 × 3 × 5 and is a multipleprime with three distinct primes. The number 60 equals 2^2 × 3 × 5; here Omega(60) = 3, so it is also a multipleprime by the multiplicity criterion, while it has three distinct primes (omega(60) = 3). Numbers such as 8 = 2^3 or 90 = 2 × 3^2 × 5 likewise qualify under Omega.

In cryptography, the term appears in reference to multiprime RSA, where the modulus n is the product of more than two primes (often three or more). Such moduli can enable faster decryption using the Chinese Remainder Theorem, at the cost of additional key-generation considerations and potential side-channel concerns. The security of multiprime schemes remains tied to the difficulty of factoring the modulus, just as in standard RSA.

See also: semiprime, RSA, Chinese Remainder Theorem.