How can non-nothing-up-my-sleeve numbers be used to exploit an algorithm?
It could be, for example, that this $N$ is a specially crafter prime number, that looks random and safe according to the NIST recommendations, yet discrete logarithms modulo $N$ can be solved considerably more efficiently due to its hidden structure. An attack of this type has been demonstrated last year at Eurocrypt in this paper; it uses a random-looking prime with a hidden structure that makes it especially sensitive to the SNFS algorithm. The team could solve discrete logarithms with respect to a $1024$-bit prime, while the current record for 'normal' primes is 768 bits.