The set of integers modulo $N$, together with the operations of both addition and multiplication forms a ring $\mathbb{Z}/ N\mathbb{Z}$. Fundamentally, this means that adding or multiplying any two elements in the set returns another element in the set.

When the modulus is prime: $N = p$, we are additionally guaranteed a multiplicative inverse of every element in the set, and so the ring is promoted to a field. In particular, we refer to this field as a finite field denoted $\Fp$.

The Diffie-Hellman protocol works with elements of some finite field $\Fp$, where the prime modulus is typically very large (thousands of bits), but for the following challenges we will keep numbers smaller for compactness.

Given the prime $p = 991$, and the element $g = 209$, find the inverse element $d = g^{-1}$ such that $g \cdot d \mod 991 = 1$.