For RSA cryptography, we know that the modulo $n$ is a product of two big prime numbers(say $p$ and $q$). However, in some documents I see an extension of $p=2p'+1$ and $q=2q'+1$ with $q'$ and $p'$ being huge prime numbers.
Why is it necessary to have $n=pq$ with $p=2p'+1$ and $q=2q'+1$ instead of just $n=p'q'$ when we know that $p'$ and $q'$ are already prime numbers?
Could you please also provide any documents/resources, if available?
In RSA as usually practiced (encryption or signature per PKCS#1, signature per X9.31, ISO/IEC 9796-2, FIPS 186), it is NOT necessary, or even common, to require $n=p⋅q$ with $p=2⋅p′+1$ and $q=2⋅q′+1$ with $p'$ and $q'$ huge primes, as stated in the question. IF that's done, it ensures that:
In RSA as usually practiced, it is generally enough to use huge random primes for $p$ and $q$. However, for relatively small modulus it is customary to require that $p-1$, $q-1$, $p+1$, and $q+1$ have one big prime factor, in order to ensure that Pollard's $p−1$ and William's $p+1$ factoring algorithms are less efficient than ECM; that precaution was mandated by ANSI X9.31, and is still in FIPS 186-4 for all $1024$ bit modulus and some prime generation methods; I believe (against the advice of many) that it remains useful when one considers an adversary content with factoring one public modulus among many.
As pointed by DrLecter, RSA-like cryptosystems or protocols using a key similar to RSA may have additional requirements; see his answer.
Actually, there are also other reasons why one wants to use safe primes in the RSA setting (when working with hidden order groups in cryptographic protocols).
When choosing the RSA modulus $n=pq$ to be the product of safe primes $p=2p'+1$ and $q=2q'+1$, then we also have the following:
The subgroup of $Z_n^*$ of qadratic residues is cyclic and has order $p'q'$. Furthermore, finding a generator of this subgroup is easy, i.e., randomly sample $h$ from $Z_n^*$, then compute $g=h^2$ (which gives us an quadratic residue by definition) and test if $\gcd(g-1,N)=1$ (the latter is proven here). If the test holds, then we have a generator of the subgroup of quadratic residues.
Note that $Z_n^*$ is not cyclic, is of unknown order (if the factorization is unknown) and it is not easy to efficiently sample elements of large order. However, when choosing the setting mentioned above we have a cyclic subgroup of large order and can efficiently sample generators for it.
Safe primes (that are two times a prime plus one) and strong primes were at some point in time considered sensible. One reason was that safe primes ensures that Pollard's $p-1$ factoring algorithm stops working. However, safe primes are not enough. There are other related factoring algorithms, such as the $p+1$ method, and strong primes also stop them.
The size of the primes used today ensures that these algorithms won't work anyway. And modern factoring algorithms (elliptic curve method, general number field sieve) work equally well for safe and strong primes. So nowadays, unless you need $n$ to have special properties (which typically means you aren't using $n$ for RSA, but for some other purpose) or need to follow a standard that for some reason still has these requirements, we use random primes instead of strong primes, since this is easier and slightly quicker.
The $p = 2p' + 1$ refers to safe primes as related to strong primes and enhances the difficulty of the discrete-log problem. This makes for a more secure system since they are more difficult to factor.
It's like a prime on top of a prime, etc…
