multiprime

Multiprime is a term used in number theory and cryptography to describe a composite integer that is the product of multiple prime factors. In general mathematics, a multiprime can mean a number that has at least three prime factors when counted with multiplicity, though in practice many authors use "multiprime" to denote a modulus that is the product of several distinct primes, commonly three or more.

In mathematical context, if the prime factors are distinct, the number is squarefree and has exactly k prime factors. If repeated factors are allowed, the same n may have k' prime factors with multiplicity. The term is distinct from semiprime (two primes) and from higher-order prime powers.

In cryptography, multi-prime RSA is a variant of RSA where the modulus n is the product of more than two primes: n = p1 p2 ... pk with k ≥ 3. The primes are chosen with sizes that sum to the desired modulus length, and the private key is derived using the Chinese remainder theorem across all primes.

Decryption and signing can be faster in multi-prime RSA because modular exponentiation is performed modulo several smaller primes and then recombined with CRT. However, generating such moduli and defending against side-channel or fault attacks requires careful parameter choices and implementation.

See also: prime factorization, RSA cryptosystem, Chinese remainder theorem, squarefree number, semiprime.