# Can multi-prime RSA be used to create an abuse-resistant lawful interception mechanism?

New to site so this may have been asked before: Can multi-prime RSA, i.e. where N is product of three or more distinct primes, be used for secure communication while allowing distinct authoritative entities to decrypt the messages when under a proper court order to do so?

For instance, let $N=p\cdot q\cdot r$, let $(N,e)$ be the public key and let $(N,d)$ the private key as usual. But have the generating algorithm securely send $(N,e,p)$ to, say, the Supreme Court, $(N,e,q)$ to the Executive and $(N,e,r)$ to Congress. Then no one entity alone could decrypt, but any pair could. Can this or some variation work? Would this require that two-prime RSA be outlawed?

It is also known that multi-prime RSA has a narrower range of exponents vulnerable to the lattice-based reduction algorithm attack. See N. Ojha and S. Padhye "Cryptanalysis of Multi Prime RSA with Secret Key Greater than Public Key", IJNS, Jan. 2014.

• The system can be secure only if each of $pq$, $qr$, and $rp$ is wide enough to be hard to factor; so if you want to be as secure a 4096-bit RSA, you need a 6144-bit public modulus.