  This article proposes a new method to inject backdoors in RSA and other
cryptographic primitives based on the Integer Factorization problem for
balanced semi-primes. The method relies on mathematical congruences among the
factors of the semi-primes modulo a large prime number, which acts as a
"designer key" or "escrow key". In particular, two different backdoors are
proposed, one targeting a single semi-prime and the other one a pair of
semi-primes. The article also describes the results of tests performed on a
SageMath implementation of the backdoors.
