In the key generation step of paillier cryptosystem ,
In order to satisfy $\gcd(pq,(p-1)(q-1))=1$ , we can take equal length primes.
Instead of taking(length as parameter to generate $p,q$) equal length primes , is there any time efficient method to generate $p,q$ that assures $pq,(p-1)(q-1)$ are relatively prime ? Or is it mandatory to take equal length primes for efficient implementation .
Just out of curiosity I want to know whether there are any public-key cryptosystems in which primes of equal length are mandatory. If there is , please list names of few of them.
Some work :
Let $p,q$ be the primes such that $p<q$.
Then $p\not|(p-1)$ and $q\not|(q-1)$
Since $p<q$ , $q\not|(p-1)$
So , the only chance to violate the condition $\gcd(pq,(p-1)(q-1))=1$ is $p|(q-1)$
More generally largest prime number minus one does not consists of smallest prime number as a prime factor .
How to ensure this without taking equal length primes ?