In RSA algorithm, given p = 7 and q = 13, find the private key.

Understand the Problem

The question is asking us to find the private key in the RSA algorithm given two prime numbers, p and q. To solve this, we will need to calculate the modulus n, the totient phi(n), and then find the private exponent d using the extended Euclidean algorithm.

Answer

The precise value of the private key $d$ will depend on the specific prime numbers $p$, $q$, and the chosen public exponent $e$.

Steps to Solve

Calculate the Modulus n

First, we need to calculate the modulus $n$ by multiplying the two prime numbers $p$ and $q$.

$$ n = p \times q $$

Calculate the Totient phi(n)

Next, we will compute the totient $\phi(n)$ using the formula:

$$ \phi(n) = (p - 1) \times (q - 1) $$

Choose the Public Exponent e

Now, we need to choose a public exponent $e$ that is coprime to $\phi(n)$. A common choice is $e = 65537$.

Find the Private Exponent d

Next, we calculate the private exponent $d$, which is the modular multiplicative inverse of $e$ modulo $\phi(n)$. We can use the Extended Euclidean Algorithm for this.

We want to solve for $d$ in the following equation:

$$ d \times e \equiv 1 \mod \phi(n) $$

Conclusion

Finally, we will have our private key $d$ after completing the calculations.

The private key $d$ can be calculated using specific values for $p$, $q$, and $e$ through the steps outlined above.

More Information

In the RSA algorithm, the private key is essential for decrypting messages that have been encrypted with the public key, which is derived from $n$ and $e$. The security of RSA is based on the difficulty of factoring the product of two large prime numbers.

Tips

Choosing a public exponent $e$ that is not coprime to $\phi(n)$ can lead to invalid results.
Forgetting to apply the Extended Euclidean Algorithm correctly may result in a wrong value for $d$.

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