Now use the RSA cryptosystem to send and receive secret messages. Working with your teacher, form teams with your classmates. Choose a name for your team. Your team should choose one of the public encryption keys listed in the public-key directory shown below. Write your team name next to your public key on a copy of the Public-Key Directory. Your teacher will hand each group their corresponding private key. Thus, each team has its own pair of keys-a public key and the associated private key.

Public-Key Directory

$$\begin{array}{|c|c|c|} \hline \text { Team Name } & \boldsymbol{n} & e \\ \hline & 85 & 13 \\ \hline & 55 & 23 \\ \hline & 161 & 19 \\ \hline & 95 & 31 \\ \hline & 91 & 29 \\ \hline & 65 & 29 \\ \hline & 35 & 13 \\ \hline \end{array}$$

a. Your team should use the RSA cryptosystem to encrypt and send at least one single-word secret message to another team. As per the cryptosystem rules, use the receiving team's public key to encrypt the message.

b. Decrypt any messages you receive. As per the cryptosystem rules, use your own private key to decrypt the message you receive.

c. Based on this public-key cryptosystem, discuss and answer the following questions.

i. Can any team send any other team a secret encrypted message? What information do you need to encrypt a secret message so that only the target team can read it?

ii. Once you encrypt a secret message to a target team, can any other team decrypt it and read the message? Explain.

iii. Suppose you receive an encrypted message from another team. What information did that team need to encrypt the message? What information do you need to decrypt it? Can any other team decrypt the message?

The ciphers in the earlier problems are called substitution ciphers because of the substitution technique used. A cipher is an algorithm used for encryption and decryption. Generally, there are two related algorithms, one for encryption and one for decryption. A substitution cipher is an algorithm that encrypts a message by replacing each character in the plaintext with another character. The receiver reverses the substitution to decrypt the message. A key typically consists of the information needed to carry out the encryption (or decryption) process. In a shift substitution cipher, a key is typically a number. For example, the key in a Caesar cipher is the number of places down the alphabet that you must shift.

a. What is the key for the Caesar cipher in the earlier problem?

b. What is the key for the ROT13 cipher?

c. Substitution ciphers like ROT13 and Caesar ciphers are not very secure. That is, it is not very difficult to decrypt the ciphertext even if you do not know the key. Describe some general strategies you think could be used to break substitution ciphers like those above.

d. Create your own substitution cipher. Encrypt a brief message. Then exchange encrypted messages with a classmate and try to decrypt each other's messages. Discuss and resolve any difficulties in the decryption process.

Now explore some other modular arithmetic systems.

a. Consider a mod 33 system. Perform the same computations as in the earlier ptoblem, except this time uses mod 33.

b. Consider a mod 15 system.

i. Why does it make sense to say that 23 and 38 are "equivalent mod 15"?

ii. What is 6$^4$ mod 15?

Examine the conjectures below-relating complements of sets.

a. Draw Venn diagrams that provide a "proof without words" for whether each conjecture is true or false.

• Conjecture I: $\quad$ (A $\cap$ B)$^{\prime}$ = A$^{\prime} \cap$ B$^{\prime}$

• Conjecture II: $\quad$ (A $\cap$ B)$^{\prime}$ = A$^{\prime} \cup$ B$^{\prime}$

b. Write and test a conjecture, similar to those above, that begins (A $\cup$ B)$^{\prime}$ = _____________. Compare your conjecture with those of your classmates. Resolve any differences.

c. For each true conjecture in Parts a and b, write the conjecture in words.

Based on your work in the earlier exercise, you could compress text data by encoding A with a shorter code word than B. To do so, you need a variable-length code. For example, consider the code below.

Variable-Length Code I

$$\begin{array}{|c|c|} \hline \text { Character } & \text { Code Word } \\ \hline \text { A } & 0 \\ \hline \text { E } & 1 \\ \hline \text { B } & 10 \\ \hline \text { C } & 11 \\ \hline \end{array}$$

a. Using this code, encode the message CAB.

b. Using this code, decode the string 0110. Compare your decoding with that of others. Resolve any differences.

c. A code is called uniquely decodable if every encoded message can be decoded to just one unique original message. Do you think this code is uniquely decodable? Why?

A standardized identification number system is used for books around the world. Each book is assigned a unique code-an International Standard Book Number (ISBN). These codes allow books to be more accurately identified and more effectively organized, especially with computerized recordkeeping. From 1970 to 2007, ISBNs had 10 digits. Since January 1, 2007, ISBNs contain 13 digits, so the numbering system is sometimes called ISBN-13. In this task and in Extensions Task 14, you will consider the older ISBN system. See Extensions Task 15 for an analysis of the ISBN-13 system.

Consider ISBN 0-380-00832-7. A 0 or 1 in the first position indicates that a book was published in an English-speaking country. The next block of digits identifies the publisher. The third block is assigned by the publisher from among the block of numbers the Library of Congress gives the publisher and it identifies the specific book. These first three blocks of digits can be of different lengths, depending on the publisher of the book. The last digit is a check digit. Here is how the check digit works.

There are ten digits in an ISBN: abcdefghij.

Compute 10a + 9b + 8c + 7d + 6e + 5f + 4g + 3h + 2i + j.

If the number is valid, this sum will be equivalent to 0 mod 11.

a. Verify that the ISBN given above is valid.

b. Consider this code: 0-88385-423-0. Is this a valid ISBN? Enter the number into the Search feature of a bookseller's Web site. What happens? Calculate to see if the ISBN is valid.

Rewrite each expression in standard polynomial form.

a. 3x(4 - 2x) + 8(x$^2$ - 5x)

b. (x - 5)(4x$^2$ + 7x - 9)

c. (5x$^3$ - 7x$^2$ + 5) - (x$^3$ +10x - 12)

d. $\frac{4 x^3-6 x^2+2 x}{2 x}$

Solve each equation.

a. 2t + 4(5t + 9) = 3

b. 8y(y - 2) + 4(6y) = 4(2y + 1.5y + y)

c. -0.15(p + 3) = p(0.75p -1.25) - 0.75p$^2$

d. $\frac{3}{8} x-\frac{1}{4}\left(\frac{1}{2} x+4\right)=\frac{3}{2}$

Rewrite each product as a single algebraic fraction. Then simplify the result as much as possible.

a. $\left(\frac{2 x^2}{9}\right)\left(\frac{12}{8 x}\right)$

b. $\left(\frac{3 a}{12}\right)\left(\frac{a+4}{a^2}\right)$

c. $\left(\frac{y^2-16}{y+1}\right)\left(\frac{y}{y+4}\right)$

In the diagram below, $\overleftrightarrow{A B} \| \overleftrightarrow{D E}$

a. Explain why $\triangle$ CED $\sim \triangle$ CBA.

b. Find the lengths of $\overline{A C}$ and $\overline{D E}$.

Fifty randomly selected families in a neighborhood in a large city were surveyed and each family was asked how many vehicles were registered at that address. The results of the survey are summarized in the table below.

$$\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Vehicles } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Families } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Families } \end{array} \\ \hline 0 & 17 & \\ \hline 1 & 23 & \\ \hline 2 & 7 & \\ \hline 3 & 2 & \\ \hline 4 & 1 & \\ \hline \end{array}$$

a. Explain what the 23 in the table indicates.

b. Use the data in the table to compute the average number of vehicles registered per address in this neighborhood.

c. Determine the percentage of families in the neighborhood that had no vehicles registered at their address. Then fill in the rest of that column of the table with percentages. Enter each percentage as a decimal number between 0 and 1. The percentage of times that a data value occurs is also called the relative frequency.

d. You can also find the average number of vehicles per address using the relative frequencies. In the formula below, f(x) is the relative frequency of the data value x. Use this formula to find the average number of vehicles registered per address.

Mean $=\sum(f(x) \cdot x)$, where x takes on all possible data values.

Your answer here should match the one you got in Part b.

e. What is the median number of vehicles registered per address in this neighborhood?

f. Which measure of center is most appropriate for these data? Explain your reasoning.

Consider the following numbers:

$$\begin{array}{lllllllll} 123 & \frac{2}{3} & \pi & -\frac{8}{4} & \sqrt[3]{8} & 31.72 & -53 & 6 \frac{3}{8} & \sqrt{7} \end{array}$$

a. Which of the numbers are whole numbers?

b. Which of the numbers are integers?

c. Which of the numbers are rational numbers?

d. Which of the numbers is irrational numbers?

e. Order the numbers from smallest to greatest.

Write each expression in a simpler equivalent exponential form using only positive exponents.

a. (3x$^6$)$^2$

b. (3$x^6$)(5x$^2$)

c. $\frac{10 x^6}{4 x^2}$

d. $\frac{-2 x^2}{8 x^6}$

e. 14(7x)$^{-1}$

f. (3x$^{-6}$)(2x$^4$)

A survey of 648 residents of Summit found that 480 of them thought the town should have a 4th of July parade.

a. What is the parameter that this survey was trying to estimate?

b. What is the point estimate of the proportion of residents of Summit who think the town should have a 4th of July parade?

c. Compute the margin of error for this survey. Then write a sentence explaining what it means in this context.