Diffie-Hellman Algorithm Overview

The Diffie-Hellman key exchange is a method for two parties to establish a shared secret key over an insecure network. It utilizes public parameters and private keys for calculation.

Diffie-Hellman Calculation Steps

The required parameters are:

• Prime modulus: $p = 47$
• Generator: $g = 3$
• A's private key: $x = 8$
• B's private key: $y = 10$

The core of the Diffie-Hellman calculation is to find the shared secret key $S$, which is computed as $S = g^{xy} \pmod{p}$.

Compute the Exponent

First, multiply the private keys selected by A and B.

$xy = 8 \times 10 = 80$

Calculate the Shared Secret

Next, raise the generator $g$ to the power of the computed exponent $(xy)$, and find the result modulo $p$.

$S = g^{xy} \pmod{p}$

$S = 3^{80} \pmod{47}$

Following the calculation using the provided parameters, the resulting shared secret value is 3.

Since Diffie-Hellman generates a single shared secret key, both participants A and B will compute the same value.

Thus, the secret keys for A and B are respectively 3 and 3.