Scenario: You are working as a security analyst for a company that needs to implement a secure communication system. Your task is to use the principles of number theory and cryptography to encode and decode messages securely$.$

Problem: The company wants to use hexadecimal to communicate, but the messages need to be encoded in binary for secure transmission. Additionally, large numbers need to be handled efficiently using modular exponentiation to ensure that encryption and decryption are fast and secure$.$

Your goal is to help the company develop a method for converting messages between different numeral systems and apply cryptographic techniques to secure the messages.

Part $1$: Base Conversion

Take the following message, written in hexadecimal: $3$E$8$A$7$D$.$

Convert this hexadecimal message into binary.

Convert the binary message into its equivalent decimal representation.

Question $1$

The hexadecimal number $3$E$8$A$7$D converts to a binary sequence. What is the $12$th bit counting from the LSB in this binary representation?