# What is multi-prime RSA (RSA-MP)?

I found some related question but no real explanation of what it is and when and why to use it. What are the benefits and downsides and is it recommended?

• possible duplicate of RSA with composite numbers Commented Feb 4, 2019 at 8:44
• Possible duplicate of Are there any standards of multi-prime RSA key generation? Commented Feb 4, 2019 at 8:45
• It's history and some benefits are also discussed here: crypto.stackexchange.com/questions/14552/… Commented Feb 4, 2019 at 8:47
• Thank you! I had only found one of those question so far! Commented Feb 4, 2019 at 9:09
• @Erwin Is your question answered by any of these Q/A's? If yes, which one. If no, what's missing? Commented Feb 5, 2019 at 1:11

Multi-prime RSA is simply using more than 2 prime numbers in generating RSA public key - The public modulus would have more than 2 factors.

We use it because it has more efficient key-generation and decryption/signing operation, which is the benefit of it.

The downside being it might be easier to factor a multi-prime RSA public key than a dual-prime one.

It's neither recommended or recommended against, it's simply an possible option without any endorsement.

• note that the primes must be distinct. Commented Feb 4, 2019 at 8:51

The security of Multi-prime RSA is analyzed in Jason Hinek's PhD Thesis which can be accessed at

In this thesis, Hinek writes

...Considering all of the known attacks on multi-prime RSA (with or without CRT decryption), the evidence suggests that multi-prime RSA with a safe number of primes is no less secure than RSA...

• With "safe number of primes" essentially defined as "no less secure than (biprime) RSA", that conclusion is certainly true!
– fgrieu
Commented Jun 7, 2022 at 11:38
• @fgrieu By "safe", he's referring back to the rest of the paper, esp. Tables 5.1 and 7.2; it's not "only" trivially true. Commented Aug 31, 2022 at 21:16
• @JamesTheAwesomeDude: yes, table 5.1 is relevant. It's a (too) much shortened version of table 1 from cited work, correctly summarized. Indeed, if referring to this, the sentence is more than a tautology. Table 7.2 is about dual RSA, thus irrelevant to that citation.
– fgrieu
Commented Sep 1, 2022 at 5:33