Blum integer

Given n = p × q a Blum integer, Q_n the set of all quadratic residues modulo n and coprime to n and a ∈ Q_n. Then:^[2]

• a has four square roots modulo n, exactly one of which is also in Q_n
• The unique square root of a in Q_n is called the principal square root of a modulo n
• The function f : Q_n → Q_n defined by f(x) = x² mod n is a permutation. The inverse function of f is: f⁻¹(x) = x^{((p−1)(q−1)+4)/8} mod n.^[4]
• For every Blum integer n, −1 has a Jacobi symbol mod n of +1, although −1 is not a quadratic residue of n:

${\displaystyle \left({\frac {-1}{n}}\right)=\left({\frac {-1}{p}}\right)\left({\frac {-1}{q}}\right)=(-1)^{2}=1}$

No Blum integer is the sum of two squares.

Before modern factoring algorithms, such as MPQS and NFS, were developed, it was thought to be useful to select Blum integers as RSA moduli. This is no longer regarded as a useful precaution, since MPQS and NFS are able to factor Blum integers with the same ease as RSA moduli constructed from randomly selected primes.^{[citation needed]}