To solve the discrete logarithm $\log_{627}(608)$ modulo a prime $p = 941$, we need to find an integer $x$ such that:

$627^x \equiv 608 \ (\text{mod} \ 941)$

This is a classical discrete logarithm problem. Solving it involves finding the exponent $x$, and one of the methods to solve it is the baby-step giant-step algorithm, a time-efficient algorithm for discrete logarithms.

Steps to solve the discrete logarithm:

1. Set the parameters:

   • The prime $p = 941$, base $g = 627$, and the result $y = 608$.
   • We aim to solve $g^x \equiv y \ (\text{mod} \ 941)$.

2. Apply the baby-step giant-step method:

   • Let $m = \lceil \sqrt{941} \rceil \approx 31$.
   • Break the problem into two steps: computing baby steps and giant steps.

Baby-step phase:

We compute and store all powers of the base $627$ for the first $m$ values:

$\text{Baby steps: } \{627^j \mod 941 \mid 0 \leq j < 31\}$

Giant-step phase:

Now, for the giant steps, compute the modular inverses of powers of $627^{-m} \mod 941$ and compare to the baby-step list.

Compute and compare:

The first value of $x$ where the two lists match gives the solution.

Using these steps, we find the discrete logarithm $x$. Let me calculate this for you!The solution to the discrete logarithm $\log_{627}(608)$ modulo $941$ is $x = 18$. This means:

$627^{18} \equiv 608 \ (\text{mod} \ 941)$

Would you like a detailed breakdown of the baby-step giant-step method used, or do you have any further questions?

Here are 5 related questions you might explore:

1. How is the baby-step giant-step algorithm derived?
2. Why is Fermat's Little Theorem used to compute modular inverses?
3. Can the discrete logarithm problem be solved efficiently for larger primes?
4. What are some real-world applications of discrete logarithms in cryptography?
5. How does the time complexity of the baby-step giant-step algorithm compare to brute force?

Tip: Modular arithmetic is widely used in cryptography, especially for encryption schemes like RSA and Diffie-Hellman key exchange.