Part $1.$

Problem $1.$ Write a C$++$ function that inputs a positive integer and determines whether it is prime

using trial division.

Use your function from Problem $1$ to write a C$++$ program for solving each of the following

computational problems. For each problem, find the running time function of the algorithm you are

using to write the corresponding program.

Problem $2.$ Determine whether $2^p-1$ is prime for each of the primes not exceeding $100.$

Problem $3.$ Find as many primes of the form $n^2+1,$ where $n$ is a positive integer, as you can.

Note: Problems $2$ and $3$ are not related to each other. You need to develop a separate program for each

problem. Your programs should invoke your function from Problem $1.$

For Part $1,$ submit your C$++$ programs, test results, and make a conclusion.

Part $2.$

Problem $4.$ Read Section $4\mathrm{.}6$ from the textbook. Give a short description of how encryption$/$decryption

of text messages can be done using a shift cipher.

Problem $5.$

a$)$ Write a $C++$ function that inputs a text message and a positive integer $k$ less than $26$ and outputs an

encrypted version of the message using the shift cipher with key $k.$

b$)$ Write a $C++$ function that inputs a positive integer $k$ less than $26$ and a text message encrypted

using the shift cipher with key $k$ and outputs a decrypted version of the message.

c$)$ Write a C$++$ program to test your functions from parts a$)$ and b$).$

Note: While working on this problem, assume that each text message consists of capital letters and

spaces only.

For Problem $5,$ submit your C$++$ program and test results. Make sure to test your program several

times.Part $1.$

Problem $1.$ Write a C$++$ function that inputs a positive integer and determines whether it is prime

using trial division.

Use your function from Problem $1$ to write a C$++$ program for solving each of the following

computational problems. For each problem, find the running time function of the algorithm you are

using to write the corresponding program.

Problem $2.$ Determine whether $2$

$1$ is prime for each of the primes not exceeding $100.$

Problem $3.$ Find as many primes of the form $$

$2+1,$ where n is a positive integer, as you can.

Note: Problems $2$ and $3$ are not related to each other. You need to develop a separate program for each

problem. Your programs should invoke your function from Problem $1.$

For Part $1,$ submit your C$++$ programs, test results, and make a conclusion.

Part $2.$

Problem $4.$ Read Section $4\mathrm{.}6$ from the textbook. Give a short description of how encryption$/$decryption

of text messages can be done using a shift cipher.

Problem $5.$

a$)$ Write a C$++$ function that inputs a text message and a positive integer k less than $26$ and outputs an

encrypted version of the message using the shift cipher with key k$.$

b$)$ Write a C$++$ function that inputs a positive integer k less than $26$ and a text message encrypted

using the shift cipher with key k and outputs a decrypted version of the message.

c$)$ Write a C$++$ program to test your functions from parts a$)$ and b$).$

Note: While working on this problem, assume that each text message consists of capital letters and

spaces only.

For Problem $5,$ submit your C$++$ program and test results. Make sure to test your program several

times.