# Cryptography with general RSA type integers?

Denote $\mathcal N_r=\{n\in\mathbb Z:\exists\mbox{ distinct equal bit primes }p_1,\dots,p_r:n=p_1p_2\dots p_{r-1}p_r\}$.

$\mathcal N_1$ refers to primes and $\mathcal N_2$ referes to balanced semiprimes used in $RSA$.

1. Does $\mathcal N_{2^t}$ at $t\in\mathbb N_{>1}$ help beyond single user-single receiver secret channel establishment?

2. Is there good references?

• Does it help ... what, exactly? The question is not clear. – Gerry Myerson Aug 7 '18 at 22:44
• @GerryMyerson Is there a multiparty protocol? In Diffie-Hellman extending to three party is straightforward. – 1.. Aug 8 '18 at 4:31
• i don't know why answer below is getting upvote. This answer was pulled up from crypto.stackexchange.com/questions/11287/…. – 1.. Aug 8 '18 at 8:34
• 'Having more than two prime factors is already supported by the PKCS#1 standard. This is called a "multiprime RSA" algorithm. On the plus size, this may offer some computational performance improvement via the Chinese Remainder Theorem. For instance, if you use a modulus with $k$ factors, the CRT speedup factor is about $k^2.$ However, using too small factors may weaken the modulus. – 1.. Aug 8 '18 at 8:35
• The best known factorization algorithms depend only on the modulus unless the factors are small enough to enter the range feasible with Elliptic Curve Multiplication which has a cost which depends (mostly) on the size of the smallest factor. More generally, a batch RSA algorithm (not multiprime RSA) can be used to speed up batch processing of many RSA signatures at once in a server setting, with substantial speedups. The paper by Boneh and Scacham [here] describes these ideas. : hovav.net/ucsd/dist/survey.pdf' – 1.. Aug 8 '18 at 8:35

On the plus size, this may offer some computational performance improvement via the Chinese Remainder Theorem. For instance, if you use a modulus with $k$ factors, the CRT speedup factor is about $k^2.$
• What is unclear here? It says 'help beyond single user-single receiver secret channel establishment?' (the context is clear $\mathcal N_2$ helps single user-single receiver channel establishment which has been known at least since $70$s as 'the RSA' algorithm). – 1.. Aug 8 '18 at 8:52