To find the value of $d$, you need to compute the modular multiplicative inverse of $e$ modulo $\phi(n)$, where $n = p \cdot q$ and $\phi(n) = (p-1)(q-1)$.

Given:

• $p = 23$
• $q = 29$
• $e = 67$

First, calculate $n$ and $\phi(n)$:

$n = p \cdot q = 23 \cdot 29 = 667$

$\phi(n) = (p-1)(q-1) = (23-1)(29-1) = 22 \cdot 28 = 616$

Now, we need to find $d$ such that:

$e \cdot d \equiv 1 \pmod{\phi(n)}$

or

$67 \cdot d \equiv 1 \pmod{616}$

We can use the Extended Euclidean Algorithm to find $d$.

1. Apply the Euclidean Algorithm to find the gcd of 67 and 616: [ \begin{align*} 616 &= 67 \cdot 9 + 13 \ 67 &= 13 \cdot 5 + 2 \ 13 &= 2 \cdot