Answer
See the explanation
Work Step by Step
To find the private keys for the RSA algorithm, we need to calculate \( d \), the private exponent, which is the multiplicative inverse of \( e \) modulo \( \phi(n) \), where \( \phi(n) \) is Euler's totient function.
Given \( n = 3233 \) and \( e = 17 \), we first need to calculate \( \phi(n) \). Since \( n \) is the product of two prime numbers \( p \) and \( q \), \( \phi(n) = (p-1)(q-1) \).
We can factorize \( n \) to find \( p \) and \( q \).
\( 3233 = 53 \times 61 \)
Therefore, \( p = 53 \) and \( q = 61 \).
Now, we calculate \( \phi(n) = (p-1)(q-1) = (53-1)(61-1) = 52 \times 60 = 3120 \).
To find \( d \), the modular multiplicative inverse of \( e \) modulo \( \phi(n) \), we can use the Extended Euclidean Algorithm or other methods.
Given that \( e = 17 \), we can find \( d \) such that \( d \cdot e \equiv 1 \mod \phi(n) \).
Calculating \( d \) using the Extended Euclidean Algorithm:
\[
17d \equiv 1 \mod 3120
\]
Solving this equation will give us \( d \).
$d=2753$
Yes, this problem can be solved in a reasonable amount of time because $n$ has a small value.
