Significance of having remainder $3$ when divided by $4$ for both $p$ and $q$ in BBS

Question

In the Blum Blum Shub random number generator, we take two random prime numbers $p$ and $q$ such that both have a remainder of $3$ when divided by $4$. My question is why can't we just take any $2$ random primes? What is significance of having remainder $3$ when divided by $4$ from the perspective of mathematics and security?

Answer 1

This is in order to maximise the state space of the generator after multiple steps. After $s$ steps, the BBS generator will have state $i^{2^s}\mod N$ where $i$ is the initial seed and $N$ is the modulus.

In particular then, the state must be $2^s$th power modulo $N$. The number of residues modulo $N$ that are $2^s$th powers is the product of the number of $2^s$th powers modulo $p$ times by the number of $2^s$th powers modulo $q$ and so we would like to maximise both of these.

The number of distinct $2^s$th powers modulo a prime $p$ is $(p-1)/2^{\mathrm{min}(s,k)}$ where $2^k$ is the largest power of $2$ that divides $p-1$. To minimise this we choose primes $p$ where $k=1$ (resp. $q$). These are the primes that are 3 modulo 4 and for these the number of $2^s$th powers will be $(p-1)/2$ (resp. $(q-1)/2$) and so the number of $2^s$th powers modulo $N$ will be $(p-1)(q-1)/4$.